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A risk-averse multi-item inventory problem with uncertain demand

  • Yanan Li
  • Ying LiuEmail author
Original Article
  • 353 Downloads

Abstract

The observed values of demands in real-life inventory problems are sometimes imprecise due to the lack of information and historical data, thus a growing research is committed to study the properties of risk measures in fuzzy inventory optimization problems. In this paper, a risk-averse fuzzy optimization method is adopted for the multi-item inventory problem, in which the demands are described by common possibility distributions. Firstly, three classes of fuzzy inventory optimization models are built by combining the absolute semi-deviation with expected value operator and then model analysis is given for the min-max inventory models. To make the inventory problem tractable and computable, the equivalent forms of the proposed optimization models are discussed. Subsequently, several useful absolute semi-deviation formulas are presented under triangular, trapezoidal and Erlang possibility distributions. Finally, some numerical experiments are performed to highlight the modeling idea, and the computational results demonstrate the effectiveness of the solution method.

Keywords

Inventory problem Uncertain demand Absolute semi-deviation Fuzzy optimization 

1 Introduction

Nowadays, inventory management problem is a significant issue in the Operations Research literature. Decision making in inventory problem is always accompanied by uncertainty especially when dealing with demands of new products, seasonal products, fashion goods and so on. The uncertainty usually brings the risk of losses. Therefore, with the knowledge of dealing with the risk properly, we can make more rational policies for the inventory problem.

In the classical modeling approach, an optimal inventory policy has to weigh the risk and the total expected profit with further consideration on the decision makers’ declared risk preference. There are some methods to measure risk, such as value-at-risk (VaR), conditional value-at-risk (CVaR) and semi-deviation. If decision makers in the inventory problem are risk aversion, they will tend to be more concerned about the risk of losses compared with profit, that is to say, they dislike the downside volatility. Hence, it is necessary to apply the downside risk measure to the inventory problem.

To make well balance between the risk and profit, we discuss the financial management in the inventory problem under uncertain demands with risk aversion attitude. In this study, we model a multi-item single-period inventory system by using min-max modeling and credibilistic optimization methods (Feng et al. 2015; Li et al. 2012; Liu and Liu 2017; Zhai et al. 2016), in which the financial characteristics are described by expected profit and absolute semi-deviation functions. In the optimization model, the uncertain demands follow several common possibility distributions. We aim at finding the optimal order quantity by minimizing the absolute semi-deviation risk and maximizing the profit with respect to the expected value simultaneously. Specifically, we firstly build inventory models and then discuss their equivalent models for easy calculation. Secondly, we deal with the calculation of the absolute semi-deviation when the demand follows common continuous distributions. Finally, the numerical discussion is carried out under various demand possibility distributions. The main contributions of this paper can be summarized as follows.
  • Absolute semi-deviation, as a kind of downside risk measure, is employed in the multi-item single-period inventory problems. This method lays stress on penalizing volatility with respect to expected profit in loss direction, which can better measure the level risk than other methods to a certain extent .

  • To solve the models we build, the calculating formulas of the absolute semi-deviation for the reciprocal of the demand are presented under three common possibility distributions, and some related proofs are listed. Although the formulas are proposed in the inventory problem we consider, they can be also applied to other fields when it comes to such calculations.

  • In the part of numerical experiment, the uncertain demands are assumed to obey various possibility distributions rather than the same possibility distribution, which reflects the diversity of demand distributions for the multiple products.

The structure of this paper is organized as follows. Section 2 gives an overview of related works. Section 3 builds a risk-averse bi-objective model and three equivalent single objective models for the multi-item single-period inventory problem and then presents the model analysis. Section 4 discusses the equivalent problem with mutually independent demand under absolute semi-deviation risk criterion, and deals with the computing of the absolute semi-deviation risk value under common continuous fuzzy demands distributions. Section 5 provides the numerical experiment to illustrate the proposed optimization methods. Section 6 gives the conclusions of this paper.

2 Literature review

Many research work on inventory problem has been made in the probabilistic framework, in which the uncertainty of demand or supply is characterized by the random demand or supply. For example, stemming from an actual case study of a low-cost textile company, Zied et al. (2014) developed a stochastic dynamic model. Ji et al. (2016) studied an optimal production planning for an assembly system in a single period setting in which the demand for the end-product is random. Lee et al. (2016) developed an extended economic production quantity model which coped with random demand. Lu et al. (2016) developed a general safety-stock determination approach under stochastic demand and random supply yield. Amiri-Aref et al. (2018) dealt with a multi-period location-inventory optimization problem in a multi-echelon supply chain network characterized by an uncertain demand and a multi-sourcing feature.

The observed values of demand and supply in real-world problems are sometimes imprecise or vague. Imprecise evaluations may result from unquantifiable, incomplete and non obtainable information. Under these considerations, some researchers addressed fuzzy uncertainty in inventory management. Baykasoglu and Gocken (2011) considered a fully fuzzy constrained multi-item economic order quantity (EOQ) model in which the parameters were defined as triangular fuzzy numbers. Chang and Yeh (2013) investigated the effects of the manufacturer’s refund on retailer’s unsold products for the two-echelon decentralized and centralized supply chains of a short life and returnable product with trapezoidal fuzzy demand. Based on the ranking of fuzzy numbers and optimization theory, Chen and Ho (2013) analyzed the optimal inventory policy for the single-order inventory problem with fuzzy demand and quantity discounts. A new robust optimization method for supply chain network design problem was presented by Bai and Liu (2014), and the proposed optimization method incorporated the uncertainties encountered in the manufacturing industry. Dash and Sahoo (2015) presented the optimization of a single period inventory problem, in which the demand was considered as a fuzzy random variable and the purchasing cost as a fuzzy number. Yang et al. (2015) proposed a new two-stage optimization method for multi-objective supply chain network design (MO-SCND) problem with uncertain transportation costs and uncertain customer demands, and developed two objectives for the SCND problem on the basis of risk-neutral and risk-averse criteria. A fuzzy-stochastic multi objective modeling approach was used by Bean et al. (2016) to address the problem of managing inventory in an environment characterised by uncertainty. Shaikh et al. (2018) studied a fuzzy inventory model for a deteriorating item in which the demand depends on selling price. Guo and Liu (2018) developed a new distributionally robust optimization method for single-period inventory problem, in which the uncertain market demand is characterized by generalized parametric interval-valued possibility distribution.

In recent years, the issues on risk management in inventory problem have received a lot of attention and many researchers have proposed various methods of risk measure. One risk measure method is directly gauging the losses rather than gauging the deviation levels from the expected profits. For instance, He et al. (2009) VaR method to study the inventory financing problem. Based on VaR, some investigators considered CVaR as a risk criterion. Sawik (2013) dealt with the optimal selection and protection of part suppliers and order quantity allocation in a supply chain with disruption risks. Wu et al. (2013) studied the effect of capacity uncertainty on the inventory decisions of a risk-averse single period inventory problem. Shi et al. (2019) presented a time-consistent dynamic CVaR measure and examined it in the context of a newsvendor problem. The other method is directly gauging the deviation levels from the expected profits. For example, Tekin and Ozekici (2015) followed a mean-variance approach to the single period inventory model in which the risks is considered in demand as well as supply. Since risk was inherently presented, Choi (2016) explored the multi-period risk minimization inventory models for fashion product purchasing via a mean-variance approach. By the expected utility functions method, Choi and Park (2015) studied a few dynamic risk-averse inventory models by using additive utility functions. There are also investigators taking semi-deviation as risk measure. Such as Chen et al. (2014) investigated the application of radio frequency identification technology to eliminate the misplacement problems in the supply chain, which consisted of a risk-neutral manufacturer and a risk-averse retailer.

In the previous work, researchers find out that decision makers tend to be more concerned about the risk of loss compared with profit, thus the concept of downside risk measure is employed in inventory problem. Semi-deviation as a downside risk measure penalizes volatility with respect to expected value in downside direction. It is the part of the actual profit below its expected value or a given objective function and is widely applied in many fields. For example, Liang and Park (2007) compared semi-deviation, VaR, expected shortfall and tail risk with standard deviation at the individual fund level as well as the portfolio level. Stadje (2010) presented an approach for the transition from convex risk measures and derived the limiting drivers for the semi-deviation risk measure. Li and Wu (2016) proposed a concept of the downside risk for the study of probabilistic optimal power flow with wind power integrated, in which the risk is manifested by semi-deviation.

To summarize, judging and weighing the risk and profit in inventory problem with uncertain demand, is critical for the decision makers. In the literature, studies have mainly focused on gauging the risk losses directly rather than gauging the deviation levels from the expected profits in the risk management of inventory problems. In this study, with the absolute semi-deviation risk measure (one method that gauging the deviation levels from the expected profits), we develop three novel inventory models to optimize the multi-item inventory problem for the risk averse decision maker in a fuzzy environment. The credibilistic optimization approach is employed to construct the min-max inventory models and the fuzzy possibility distributions are adopted to describe the uncertain demands.

3 Multi-item single-period inventory problem

In real life, the inventory problem is of significance in terms of both theoretical and practical consideration. The problem studied in this paper is the multi-item single-period inventory problem under risk-averse criterion. In this section, we will firstly introduce our inventory problem briefly and then try to build rational models.

3.1 Notations

Suppose a firm allow the buyers to order goods in advance, and it can obtain the revenue through the order quantity. Since the firm allows buyers ordering goods in advance, we consider two types of costs. One is the fixed order cost component and the other is the holding cost which are both paid at the beginning in the multi-item inventory problem, and we only consider one period. We aim at maximizing the profit and minimizing inventory risk. In order to describe our problem clearly, we adopt the notations in the following text.

Notations

c = [c1, c2,⋯ , cn]:

Unit fixed costs per inventoried item;

d = [d1, d2,⋯ , dn]:

Unit revenues per inventoried item;

g = [g1, g2,⋯ , gn]:

Unit holding costs per inventoried item;

ξ = [ξ1, ξ2,⋯ , ξn]:

Fuzzy demand vector in the inventory problem;

x = [x1, x2,⋯ , xn]:

Order quantity vector in the inventory problem;

γ:

Absolute semi-deviation parameter;

n:

Numbers of the inventoried item.

If decision makers want to obtain the maximum profit under risk-averse criterion, they usually use the expected value and risk-averse measure of the profit as objective functions, then maximize the total expected profit and minimize the risk simultaneously.

3.2 Formulation of models under absolute semi-deviation criterion

In this subsection, we mainly introduce the absolute semi-deviation risk criterion and discuss the formulation of models for the multi-item inventory problem.

Let us denote i = 1,2,…, n to be the types of items and suppose there is no influence between any two items. xi is the order quantity of i th item and di denotes the revenue of i th item. Therefore, dixi is the total revenue of i th item in the inventory problem. We know that ci is the fixed order cost of i th item, gi is the unit holding cost of i th item and ξi is the demand of i th item. Thus \(g_{i}{x_{i}^{2}}/2\xi _{i}\) is the holding cost of i th item in the inventory problem. If denote πi(xi, ξi) as the profit functions of item i, then we have the following expression:
$$ \pi_{i}(x_{i},\xi_{i})=d_{i}x_{i}-c_{i}-\frac{g_{i}{{x_{i}^{2}}}}{2\xi_{i}}. $$
If we denote π(x,ξ) as the total profit function of the inventory problem, then the multi-item profit function π(x,ξ) is expressed as
$$ \pi(\textbf{x},\boldsymbol{\xi})=\sum\limits_{i=1}^{n}\pi_{i}(x_{i},\xi_{i})=\sum\limits_{i=1}^{n}\left( d_{i}x_{i}-c_{i}-\frac{g_{i}{{x_{i}^{2}}}}{2\xi_{i}}\right). $$
(1)

Due to the influence of various factors, demand ξ is uncertain. Using fuzzy variable to describe the demand ξ, then the uncertain profit function π(x,ξ) is a function with respect to fuzzy demand. In order to determine the optimal policy of replenishment, we need to determine the expected parts (profit and cost) associated to each policy. The expected value of fuzzy profit π(x,ξ) is denoted by E[π(x,ξ)].

As we know, if profit is the only concern, it is simple for the decision maker to make order policy in the inventory problem with the highest expected profit. However, the high profit is usually accompanied with high level of risk. Therefore, we need to adopt a type of appropriate risk measure related to the profit π(x,ξ) to make our problem more meaningful in real life and help the decision maker make proper order policy.

Some literature for inventory problem with risk uses variance as a risk measure, such as Borgonovo and Peccati (2009) and Katariya et al. (2014). For variance criterion, it is the degree of deviation from the expected value, and is used to measure the size of a group of data volatility in both upside and downside directions. In fact, the upside of the variance can be considered as the extra gains from the inventory management. Most decision makers are more concerned about the downside losses compared with the upside gains. Thus, the absolute semi-deviation as a kind of downside risk measure occurs in the inventory problem, and represents the risk aversion attitude of decision makers. Under the risk-averse criterion, we take the absolute semi-deviation (Chen et al. 2012) as a risk measure that the decision maker can maximize the profit and minimize the risk at the same time.

Let ξ be a fuzzy variable with finite expected value. Then the absolute semi-deviation variable (Chen et al. 2012) of ξ with respect to expected value is as follows:
$$ \begin{array}{@{}rcl@{}} \qquad &(\xi-\mathrm{E}[\xi])^{-}=\left\{ \begin{array}{lllll} &\mathrm{E}[\xi]-\xi,&& \xi \leq \mathrm{E}[\xi]\\ &0,&& \xi >\mathrm{E}[\xi], \end{array} \right. \end{array} $$
and
$$ \begin{array}{@{}rcl@{}} \qquad &(\xi-\mathrm{E}[\xi])^{+}=\left\{ \begin{array}{lllll} &\xi-\mathrm{E}[\xi],&& \xi \geq \mathrm{E}[\xi]\\ &0,&& \xi <\mathrm{E}[\xi]. \end{array} \right. \end{array} $$
Let ξ be a fuzzy variable with finite expected value, and Φ(x) its credibility distribution. If L-S integral \({\int }_{(-\infty ,+\infty )}x\mathrm {d}{\Phi }(x)\) is finite, then the expected lower absolute deviation of ξ with respect to expected value is defined as the following expected value:
$$ \rho^{-}(\xi)={\int}_{(-\infty,\mathrm{E}[\xi])}(\mathrm{E}[\xi]-x)\mathrm{d}{\Phi}(x), $$
and the expected upper absolute deviation of ξ is defined as the next expected value:
$$ \rho^{+}(\xi)={\int}_{(\mathrm{E}[\xi],+\infty)}(x-\mathrm{E}[\xi])\mathrm{d}{\Phi}(x). $$

For the absolute semi-deviation criterion denotes the degree of deviation from the risk in the inventory problem, then we take the maximum of the absolute semi-deviation for the i th profit function as an objective function, and it reads that

$$ \max_{1\leq i\leq n}\left\{\rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\right\}=\max_{1\leq i\leq n}\left\{\mathrm{E}\left[\pi_{i}(x_{i},\xi_{i})-\mathrm{E}\left[\pi_{i}(x_{i},\xi_{i})\right]\right]^{-}\right\}. $$
Considering the risk-averse profit function, we aim at seeking the minimum risk and obtaining the maximum expected profit. Thus we take the expected value and the absolute semi-deviation risk criterion as two measure indexes, then the bi-objective model for the multi-item inventory problem is formally built as
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{lll} \max \quad &\mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})]\\ \min \quad &\max_{1\leq i\leq n}\left\{\rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\right\}\\ \text{s. t.}\quad &\textbf{x}\geq 0. \end{array} \right. \end{array} $$
(2)
If a decision maker is looking for the order policy with minimum risk under prescribing a minimum acceptable profit level r0 of expected ordering profit, the problem (2) can be turned into the following single objective risk minimization programming model:
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{lll} \min \quad & \max_{1\leq i\leq n}\left\{\rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\right\}\\ \text{s. t.}\quad & \mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})]\geq r_{0},\\ \quad & \textbf{x}\geq 0. \end{array} \right. \end{array} $$
(3)
On the other hand, if a decision maker desires to maximize expected profit under the condition that the maximum acceptable risk level is s0, then the problem (2) can be turned into the following single objective profit maximization programming model:
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{lll} \max \quad & \mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})]\\ \text{s. t.}\quad & \max_{1\leq i\leq n}\left\{\rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\right\}\leq s_{0},\\ \quad & \textbf{x}\geq 0. \end{array} \right. \end{array} $$
(4)
Furthermore, to consider expected profit and absolute semi-deviation risk simultaneously in the objective function, the problem (2) can be built as the following single objective programming model:
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{lllll} \max \quad & \mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})] - \gamma\max_{1\leq i\leq n}\left\{\rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\right\}\\ \text{s. t.}\quad & \textbf{x}\geq 0, \end{array} \right. \end{array} $$
(5)
where γ is a positive risk measure parameter with respect to the absolute semi-deviation. The parameter γ describes the importance of risk relative to the absolute semi-deviation, lower values of γ attempt to maximize the expected profit regardless of risk, while higher values of γ tend to minimize the risk.

3.3 Model analysis

According to the min-max modeling method, we transform models (3)–(5) into their equivalent forms, which are as follows.

For model (3), we introduce an additional variable t and let \(t=\max _{1\leq i\leq n}\left \{\rho ^{-}\left (\pi _{i}(x_{i},\xi _{i})\right )\right \}\). Thereby, model (3) can be translated to its liner objective equivalent form:
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{llll} \min \quad & t\\ \text{s. t.}\quad & \mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})]\geq r_{0},\\ \quad & \rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\leq t, \ i=1,2,\ldots,n,\\ \quad & \textbf{x}\geq 0. \end{array} \right. \end{array} $$
(6)
In model (4), the constraint \(\max _{1\leq i\leq n}\left \{\rho ^{-}\left (\pi _{i}(x_{i},\xi _{i})\right )\right \}\leq s_{0}\) holding means that for any i, i = 1,2,…, n, we have \(\rho ^{-}\left (\pi _{i}(x_{i},\xi _{i})\right )\leq s_{0}\). Then model (4) can be translated to its liner constraints equivalent form:
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{llll} \max \quad & \mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})]\\ \text{s. t.}\quad & \rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\leq s_{0},\ i=1,2,\ldots,n,\\ \quad & \textbf{x}\geq 0. \end{array} \right. \end{array} $$
(7)

For model (5), we introduce an additional variable e and let \(e=\min _{1\leq i\leq n} \left \{\mathrm {E}[\pi (\textbf {x},\boldsymbol {\xi })] - \gamma \rho ^{-}\left (\pi _{i}(x_{i},\xi _{i})\right )\right \}\). Hence, model (5) can be translated to its liner objective equivalent form:

$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{ll} \max \quad & e \\ \text{s. t.}\quad & \mathrm{E}[\pi(\textbf{x},\boldsymbol{\xi})]-\gamma\rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)\geq e, \ i=1,2,\ldots,n,\\ \quad & \textbf{x}\geq 0, \end{array} \right. \end{array} $$
(8)
where γ is a positive risk measure parameter with respect to the absolute semi-deviation.

Through analyzing the models for the multi-item inventory problem, we have gotten the equivalent models by the min-max modeling method. In next section, we will discuss the specific equivalent forms of models (6)–(8) when the demands ξi, i = 1,2,…, n are mutually independent fuzzy variables.

4 Equivalent problem under absolute semi-deviation criterion

4.1 Forms of absolute semi-deviation under independence

In order to solve models (6)–(8), the key point is to compute the expected value and absolute semi-deviation of i th profit function πi(xi, ξi). In the section, we will discuss the calculation of E[π(x,ξ)] and ρ[πi(xi, ξi)].

If we denote the demand vector ξ = (ξ1, ξ2,⋯ , ξn), and suppose the demands ξi, i = 1,2,…, n are mutually independent fuzzy variables (Liu and Gao 2007), then the joint possibility distribution μξ is represented by
$$ \mu_{\boldsymbol{\xi}}(t_{1},t_{2},\cdots,t_{n})=\min_{1\leq i\leq n}\mu_{\xi_{i}}(t_{i}). $$
As πi(xi, ξi) are fuzzy profit functions, then πi(xi, ξi), i = 1,2,…, n are also mutually independent. Due to the independence linearity of expected value operator (Liu and Liu 2003), wealth and costs can be separated from the expected equation as they do not affect the optimal ordering quantity. Then, the expected profit function is
$$ \mathrm{E}\left[\pi(\textbf{x},\boldsymbol{\xi})\right]=\sum\limits_{i=1}^{n}\mathrm{E}\left[\pi_{i}(x_{i},\xi_{i})\right], $$
and the separated result has been derived in Li and Liu (2016), which is as follows:
$$ \mathrm{E}\left[\pi(\textbf{x},\boldsymbol{\xi})\right]=\sum\limits_{i=1}^{n}\left( d_{i}x_{i}-c_{i}-\frac{g_{i}{{x_{i}^{2}}}}{2}\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right). $$
Furthermore, we have the expression of the absolute semi-deviation \(\rho ^{-}\left (\pi _{i}(x_{i},\xi _{i})\right )\) which is as follows:
$$ \rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)=\mathrm{E}\left[-\frac{g_{i}{{x_{i}^{2}}}}{2}\left( \frac{1}{\xi_{i}}-\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)\right]^{-}. $$
If ξi are mutually independent fuzzy variables and bi ∈R, i = 1,2,…, n, then we have the following conclusions:
  1. (i)
    if bi ≥ 0, then the equivalent expression is
    $$ \mathrm{E}\left[b_{i}\left( \frac{1}{\xi_{i}}-\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)\right]^{-}=b_{i}\mathrm{E}\left( \frac{1}{\xi_{i}}-\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)^{-}, $$
     
  2. (ii)
    if bi < 0, then the equivalent expression is
    $$ \mathrm{E}\left[b_{i}\left( \frac{1}{\xi_{i}}-\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)\right]^{-}=-b_{i}\mathrm{E}\left( \frac{1}{\xi_{i}}-\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)^{+}. $$
     

4.2 Equivalent forms of original models

In this subsection, we will transform the original problems (6)–(8) into their equivalent problems to facilitate the calculation and model analysis.

If bi < 0, i = 1,2,⋯ , n, then it holds that
$$ \rho^{-}\left( \pi_{i}(x_{i},\xi_{i})\right)=-b_{i}\mathrm{E}\left[\frac{1}{\xi_{i}}-\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right]^{+}. $$
Let \(b_{i}=-g_{i}{{x_{i}^{2}}}/{2}, i=1,2,\cdots ,n\). It is obvious that \(-g_{i}{{x_{i}^{2}}}/{2}< 0\), then model (6) is transformed into the following equivalent risk minimization programming model
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{lll} \min \quad & t\\ \text{s. t.}\quad & {\sum}_{i=1}^{n}\left( d_{i}x_{i}-c_{i}-\frac{g_{i}m_{i}{{x_{i}^{2}}}}{2}\right)\geq r_{0},\\ \quad & \frac{g_{i}{{x_{i}^{2}}}}{2}\mathrm{E}\left( \frac{1}{\xi_{i}}-m_{i}\right)^{+}\leq t,\ i=1,2,\ldots,n,\\ \quad & \textbf{x}\geq 0, \end{array} \right. \end{array} $$
(9)
where \(m_{i}=\mathrm {E}\left [\frac {1}{\xi _{i}}\right ],\quad i=1,2,\ldots ,n\).
Similarly, model (7) is transformed into the following equivalent profit maximization programming model
$$ \begin{array}{@{}rcl@{}} \qquad &\left\{ \begin{array}{llll} \max \quad & {\sum}_{i=1}^{n}\left( d_{i}x_{i}-c_{i}-\frac{g_{i}m_{i}{{x_{i}^{2}}}}{2}\right)\\ \text{s. t.}\quad & \frac{g_{i}{{x_{i}^{2}}}}{2}\mathrm{E}\left( \frac{1}{\xi_{i}}-m_{i}\right)^{+}\leq s_{0},\ i=1,2,\ldots,n,\\ \quad & \textbf{x}\geq 0, \end{array} \right. \end{array} $$
(10)
and model (8) is transformed into the following equivalent programming model
$$ \begin{array}{@{}rcl@{}} &\left\{ \begin{array}{llll} \max \quad & e\\ \text{s. t.}\quad & {\sum}_{i=1}^{n}\left( d_{i}x_{i}-c_{i}-\frac{g_{i}m_{i}{{x_{i}^{2}}}}{2}\right) -\gamma\frac{g_{i}{{x_{i}^{2}}}}{2}\mathrm{E}\left( \frac{1}{\xi_{i}}-m_{i}\right)^{+}\geq e,\ i=1,2,\ldots,n,\\ \quad & \textbf{x}\geq 0, \end{array} \right. \end{array} $$
(11)
where \(m_{i}=\mathrm {E}\left [\frac {1}{\xi _{i}}\right ],\quad i=1,2,\ldots ,n\).

In order to solve these models,we need to compute the values of E[1/ξi] and \(\mathrm {E}\left [\left (\frac {1}{\xi _{i}}-m_{i}\right )^{+}\right ], i=1,2,\ldots ,n\). We have obtained the expected value E[1/ξi] in Li and Liu (2016) when demand ξi, i = 1,2,…, n obey common fuzzy distributions. In the next section, we will discuss the equivalent forms of absolute semi-deviation value \(\mathrm {E}\left [\left (\frac {1}{\xi _{i}}-m_{i}\right )^{+}\right ], i=1,2,\ldots ,n\) under common demand distributions.

4.3 Computing of absolute semi-deviation

In this section, we will deal with the computing of the absolute semi-deviation value \(\mathrm {E}\left [\left (\frac {1}{\xi _{i}}-m_{i}\right )^{+}\right ],(i=1,2,\ldots ,n)\) under triangular, trapezoidal and Erlang fuzzy demand distributions.

Firstly, we discuss the related results under triangular fuzzy demand distribution, which are stated as:

Theorem 1

Assume that demandξisa triangular fuzzy variable (r1, r2, r3) withr1 > 0 andm = E[1/ξ],then the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)iscomputed by

$$ \begin{array}{@{}rcl@{}} &&\mathrm{E}\left[\left( \frac{1}{\xi}-m\right)^{+}\right]\\&&=\left\{ \begin{array}{llll} &\frac{1}{2(r_{3}-r_{2})}\ln \frac{1}{r_{2}m}+\frac{1}{2(r_{2}-r_{1})}\ln \frac{r_{2}}{r_{1}}+\frac{m(2r_{2}-r_{3})-1}{2(r_{3}-r_{2})},&& \frac{1}{r_{3}}\leq m<\frac{1}{r_{2}}\\ &\frac{1}{2(r_{2}-r_{1})}\ln \frac{1}{r_{1}m}+\frac{mr_{1}-1}{2(r_{2}-r_{1})},&& \frac{1}{r_{2}}\leq m<\frac{1}{r_{1}}. \end{array} \right.\\ \end{array} $$
(12)
Note that whenr1 = 0,the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)doesnot exist.

The proof of Theorem 1 is shown in Appendix.

Next, for the trapezoidal fuzzy demand distribution, we have related results as follows:

Theorem 2

Assume that demandξisa trapezoidal fuzzy variable (r1, r2, r3, r4) withr1 > 0 andm = E[1/ξ],then the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)iscomputed by

$$ \begin{array}{@{}rcl@{}} &&\mathrm{E}\left[\left( \frac{1}{\xi}-m\right)^{+}\right]\\ &&=\left\{ \begin{array}{llll} &\frac{1}{2(r_{4}-r_{3})}\ln \frac{1}{r_{3}m}+\frac{1}{2(r_{2}-r_{1})}\ln \frac{r_{2}}{r_{1}}+\frac{m(2r_{3}-r_{4})-1}{2(r_{4}-r_{3})},&& \frac{1}{r_{4}}\leq m<\frac{1}{r_{3}}\\ &\frac{1}{2(r_{2}-r_{1})}\ln \frac{r_{2}}{r_{1}}-\frac{m}{2},&& \frac{1}{r_{3}}\leq m<\frac{1}{r_{2}}\\ &\frac{1}{2(r_{2}-r_{1})}\ln \frac{1}{r_{1}m}+\frac{mr_{1}-1}{2(r_{2}-r_{1})},&& \frac{1}{r_{2}}\leq m<\frac{1}{r_{1}}. \end{array} \right. \end{array} $$
Note that whenr1 = 0,the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)doesnot exist.

The proof of Theorem 2 is shown in Appendix.

Finally, for the Erlang fuzzy demand distribution, we have related results as follows:

Theorem 3

Assume that demandξisan Erlang fuzzy variable Er(λ, r),whereξis variedon [r1, r2] withr1 > 0, r is a positiveinteger andλ > 0.
  1. (i)

    If\(\frac {1}{r_{2}}\leq m<\frac {1}{\lambda r}\)andr = 1,then the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)iscomputed by

    $$ \begin{array}{@{}rcl@{}} \mathrm{E}\left[\left( \frac{1}{\xi}-m\right)^{+}\right] &=& \left( \frac{1}{r_{1}} - m\right) \left( 1 - \frac{r_{1}}{2\lambda}e^{1 - \frac{r_{1}}{\lambda}}\right) - \left( \frac{1}{r_{1}} - \frac{1}{\lambda}\right)\\ && + \frac{e}{2\lambda}\left[\ln\frac{m\lambda^{2}}{r_{1}} + \sum\limits_{i=1}^{+\infty} \frac{2( - 1)^{i} - \left( - \frac{1}{\lambda m}\right)^{i} - \left( - \frac{r_{1}}{\lambda}\right)^{i}}{i\cdot i!}\right], \end{array} $$
    (13)
    where the expected valuem is computed by
    $$ m=\frac{e}{2\lambda}\left[\ln\frac{\lambda^{2}}{r_{1}r_{2}}+\sum\limits_{i=1}^{+\infty}\frac{2(-1)^{i}-\left( -\frac{r_{2}}{\lambda}\right)^{i}-\left( -\frac{r_{1}}{\lambda}\right)^{i}}{i\cdot i!}\right]+\frac{1}{\lambda}-\frac{1}{r_{2}}. $$
    (14)
     
  2. (ii)
    If\(\frac {1}{r_{2}}\leq m<\frac {1}{\lambda r}\)and r = n,(n > 1), then the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)is computed by
    $$ \begin{array}{@{}rcl@{}} \mathrm{E}[(\eta-m)^{+}]&=&\left( \frac{1}{r_{1}}-m\right)\left[1-\frac{1}{2}\left( \frac{r_{1}}{\lambda r}\right)^{r}e^{r-\frac{r_{1}}{\lambda}}\right]-\left( \frac{1}{r_{1}}-\frac{1}{\lambda r}\right)+\frac{e^{r}}{2\lambda}\left( \frac{1}{r}\right)^{r}\left[\left( \frac{1}{\lambda m}\right)^{r-2}e^{-\frac{1}{\lambda m}}\right.\\ && + \left. \left( \frac{r_{1}}{\lambda}\right)^{r-2}e^{-\frac{r_{1}}{\lambda}}-2r^{r-2}e^{-r}+(r-2)\left( {\int}_{(\frac{1}{\lambda m},r)}t^{r-3}e^{-t}\text{dt}-{\int}_{(r,\frac{r_{1}}{\lambda})}t^{r-3}e^{-t}\text{dt}\right)\right], \end{array} $$
    (15)

    where the expected value m is computed by

    $$ \begin{array}{@{}rcl@{}} m&=&\frac{e^{r}}{2\lambda}\left( \frac{1}{r}\right)^{r} \left[ - 2r^{r - 2}e^{-r} + \left( \frac{r_{2}}{\lambda}\right)^{r - 2}e^{ - \frac{r_{2}}{\lambda}} + \left( \frac{r_{1}}{\lambda}\right)^{r - 2}e^{ - \frac{r_{1}}{\lambda}}\right.\\ &&+ \left. (r - 2) \left( {\int}_{\frac{r_{2}}{\lambda}}^{r} t^{r - 3}e^{ - t}\text{dt} - {\int}_{r}^{\frac{r_{1}}{\lambda}} t^{r - 3}e^{ - t}\text{dt}\right)\right] + \frac{1}{\lambda r} - \frac{1}{r_{2}}. \end{array} $$
    (16)
     
  3. (iii)
    If\(\frac {1}{\lambda r}\leq m \leq \frac {1}{r_{1}}\)and r = 1, then the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)is computed by
    $$ \begin{array}{@{}rcl@{}} &&\mathrm{E}\left[\left( \frac{1}{\xi}-m\right)^{+}\right]\\&&=-\left( \frac{1}{r_{1}}-m\right)\frac{r_{1}e^{1-\frac{r_{1}}{\lambda}}}{2\lambda}-\frac{e}{2\lambda}\left[\ln mr_{1}+\sum\limits_{i=1}^{+\infty}\frac{\left( -\frac{r_{1}}{\lambda}\right)^{i}-\left( -\frac{1}{\lambda m}\right)^{i}}{i\cdot i!}\right], \end{array} $$
    (17)

    where the expected value m of η is computed by the formulation (14).

     
  4. (iv)

    If\(\frac {1}{\lambda r}\leq m \leq \frac {1}{r_{1}}\)and r = n,(n > 1), then the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)is computed by

    $$ \begin{array}{@{}rcl@{}} \mathrm{E}[(\eta - m)^{+}]&=& - \frac{1}{2}\left( \frac{1}{r_{1}} - m \right) \left( \frac{r_{1}}{\lambda r}\right) ^{r} e^{r - \frac{r_{1}}{\lambda}}\\&&+ \frac{e^{r}}{2\lambda} \left( \frac{1}{r}\right)^{r} \left[ \left( \frac{r_{1}}{\lambda}\right)^{r - 2}e^{ - \frac{r_{1}}{\lambda}} - \left( \frac{1}{\lambda m}\right) ^{r - 2}e^{ - \frac{1}{\lambda m}}\right.\\ && - \left. (r-2) {\int}_{\left( \frac{1}{\lambda m},\frac{r_{1}}{\lambda}\right)} t^{r - 3}e^{ - t}\text{dt} \right], \end{array} $$
    (18)
    where the expected value m of η is computed by the formulation (16).

    Note that when r1 = 0, the absolute semi-deviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }-m\right )^{+}\right ]\)does not exist.

     

The proof of Theorem 2 is shown in Appendix.

5 Numerical experiment

To assess the performance of the proposed models (9)–(11), the numerical experiment is implemented and the related results are reported in this section. Firstly, the problem description for a given factory is presented. Secondly, the numerical analysis and computational results for models (9)–(11) are provided.

5.1 Problem description

Suppose that a clothing factory produces multiple clothing and allows buyers to order those clothing in advance. In this way, it can obtain revenue through the order quantity. There are two types of costs, the fixed order cost and the holding cost for the clothing factory. The firm’s objective is to determine the optimal ordering quantity so as to optimize expected total profit under risk-averse criterion. The economics parameters d, c and g in the inventory problem are provided in Table 1, where d denotes the revenues per inventoried clothing, c denotes the fixed costs per inventoried clothing and g denotes the holding costs per inventoried clothing.
Table 1

Economics inputs for the inventory problem ($)

Item

1

2

3

4

5

6

7

8

9

10

d

10

11

12.5

13

12

9.5

14

13.5

12.5

15

c

1

2

2.5

1.5

1.8

2.2

2.3

4.1

1.9

2.7

g

0.55

0.6

0.65

0.71

0.53

0.56

0.68

0.81

0.92

0.5

5.2 Numerical analysis and computational results

In this subsection, we utilize models (9)–(11) and do some experiments under fuzzy demands which can help the decision maker of the clothing factory make optimal order policy.

Computational results of model (9)

For 10 items in the inventory problem, demands ξi(i = 1,2,3,4) obey triangular distributions \(({r_{1}^{i}},{r_{2}^{i}},{r_{3}^{i}})\), ξi(i = 5,6,7) obey trapezoidal distributions \(({r_{1}^{i}},{r_{2}^{i}},{r_{3}^{i}},{r_{4}^{i}})\) and ξi(i = 8,9,10) obey Erlang distributions Er(λi, ri), where ξi ∈ [1,40]. The possibility distributions of demands ξi, i = 1,…,10 are provided in Table 2.
Table 2

The distributions of fuzzy demands ξi

Item

Fuzzy demands ξi

Item

Fuzzy demands ξi

Item

Fuzzy demands ξi

1

(10,20,30)

5

(9,24,39,44)

8

Er(10, 2)

2

(20,30,40)

6

(10,15,25,30)

9

Er(15, 2)

3

(5,15,25)

7

(8,20,30,40)

10

Er(5, 3)

4

(15,30,45)

    
On the basis of Theorems 1–3 in this Section 4.3 and some conclusions in Li and Liu (2016), we obtain the expected value E[1/ξi]. We take \(({r_{1}^{1}},{r_{2}^{1}},{r_{3}^{1}})=(10,20,30)\) as an example to show how to compute the value of the absolute semi-deviation E[(1/ξi −E[1/ξi])+], i = 1,2,…,10. We know that E[1/ξ1] = 0.0549 in the case of \(({r_{1}^{1}},{r_{2}^{1}},{r_{3}^{1}})=(10,20,30)\). Obviously, \(1/{r_{2}^{1}}\leq 0.0549<1/{r_{1}^{1}}\). Thus, take E[(1/ξ1 −E[1/ξ1])+] into the corresponding case of (12), we can get the computational result. Similarly, other values of the absolute semi-deviation can also be obtained and the results are shown in Table 3.
Table 3

The calculation results of the expected value and absolute semi-deviation

Expected value and

Item

absolute semi-deviation

1

2

3

4

5

\(\mathrm {E}\left [\frac {1}{\xi _{i}}\right ]\)

0.0549

0.0347

0.0805

0.0366

0.0448

\(\mathrm {E}\left (\frac {1}{\xi _{i}}-\mathrm {E}\left [\frac {1}{\xi _{i}}\right ]\right )^{+}\)

0.0074

0.0030

0.0156

0.0050

0.0104

Expected value and

Item

absolute semi-deviation

6

7

8

9

10

\(\mathrm {E}\left [\frac {1}{\xi _{i}}\right ]\)

0.0588

0.0526

0.0853

0.0535

0.0853

\(\mathrm {E}\left (\frac {1}{\xi _{i}}-\mathrm {E}\left [\frac {1}{\xi _{i}}\right ]\right )^{+}\)

0.0111

0.0119

0.0473

0.0363

0.0470

Subsequently, take the results of the expected value E[1/ξi] and the absolute semi-deviation E[(1/ξi −E[1/ξi])+] into model (9), we obtain the optimal order policy x by Lingo when the minimum acceptable profit level r0 is given. The optimal order policy x under different minimum acceptable profit level r0 is shown in Table 4.
Table 4

Optimal order policy under different minimum acceptable profit levels

Minimum acceptable

Optimal order policy

profit level

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

r0 = 5000

59

90

37

63

51

48

42

19

20

24

r0 = 10000

129

193

81

138

110

104

91

42

45

53

r0 = 15000

218

328

138

234

187

177

155

71

76

91

r0 = 20000

331

530

239

416

334

285

276

126

135

161

r0 = 21970

331

528

239

500

505

290

390

194

254

345

According to the data in Tables 13 and 4, the minimum objective value \(\max _{1\leq i\leq n} \rho ^{-} [\pi (x_{i}^{\ast },\xi _{i})]\) of model (9) under different minimum acceptable profit profit levels r0 is shown in Table 5.
Table 5

The calculation results of the minimum objective function value

Minimum objective

Minimum acceptable profit level

function value($)

r0 = 5000

r0 = 10000

r0 = 15000

r0 = 20000

r0 = 21700

\(\max _{1\leq i\leq n}\rho ^{-}[\pi (x_{i}^{\ast },\xi _{i})]\)

7.29

33.86

97.37

308.21

1398.54

The numerical experiment demonstrates that model (9) can provide diversified order policies to the inventory problem. From Table 5 and Figs. 1 and 2, we can see when the minimum acceptable profit level r0 is varied, the absolute semi-deviation \(\max _{1\leq i\leq n}\rho ^{-}[\pi (x_{i}^{\ast },\xi _{i})]\) is varied accordingly. The computational results demonstrate that the higher the profits we desire for, the greater the risk we will meet. Therefore, it indicates that the model we build for the multi-item inventory problem is consistent with reality.
Fig. 1

The relationship between the absolute semi-deviation and minimum acceptable profit level r0

Fig. 2

The relationship between the expected profit and maximum acceptable risk level s0

Computational results of model (10)

The possibility distributions of demands ξi,(i = 1,2,…,10) are provided in Table 2. The expected value E[1/ξi] and the absolute semi-deviation value E[(1/ξi −E[1/ξi])+], i = 1,2,…,10 are shown in Table 3.

Subsequently, take the results of the expected value E[1/ξi] and the absolute semi-deviation E[(1/ξi −E[1/ξi])+] into model (10), we obtain the optimal order policy x by Lingo when the maximum acceptable risk level s0 is given. The optimal order policy x under different maximum acceptable risk levels s0 is listed in Table 6.
Table 6

Optimal order policy under different maximum acceptable risk levels

Maximum acceptable

Optimal order policy

risk level

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

s0 = 50

156

235

99

167

134

126

111

51

54

65

s0 = 100

221

333

140

237

190

179

157

72

77

92

s0 = 500

331

528

239

500

425

289

351

161

173

206

s0 = 1000

331

528

239

500

505

289

391

195

244

291

s0 = 1500

331

528

239

500

505

289

391

195

254

352

According to the data in Tables 13 and 6, the maximum objective value E[π(x,ξ)] of model (10) under several different maximum acceptable risk levels s0 is shown in Table 7.
Table 7

The calculation results of the maximum objective function value

Maximum objective

Maximum acceptable risk level

function value($)

s0 = 50

s0 = 100

s0 = 500

s0 = 1000

s0 = 1500

E[π(x,ξ)]

11721.29

15138.26

21210.31

21890.08

21971.09

From Table 7 and Fig. 2, we can see when the maximum acceptable risk level s0 is varied, the inventory total expected profit E[π(x,ξ)] is varied accordingly. The computational results demonstrate that the greater the risk-seeking, the higher the profits can be obtained. It is also meaningful in the real life.

Computational results of model (11)

The possibility distributions of demands ξi,(i = 1,2,…,10) are provided in Table 2. The expected value E[1/ξi] and the absolute semi-deviation value E[(1/ξi −E[1/ξi])+], i = 1,2,…,10 are shown in Table 3. The optimal order policy x with various values of risk measure parameter γ is presented in Table 8.
Table 8

Optimal order policy under different risk measure parameters

Risk measure

Optimal order policy

parameter

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

γ = 0

331

528

239

500

505

289

391

195

254

352

γ = 0.5

331

528

239

500

505

289

391

195

239

285

γ = 1

331

528

239

500

505

289

391

195

213

254

γ = 1.5

331

528

239

500

487

289

391

184

198

236

γ = 2

331

528

239

500

475

289

391

180

193

230

γ = 2.5

331

528

239

500

456

289

376

173

185

221

According to the data in Tables 13 and 8, the maximum objective function value \(\mathrm {E}[\pi (\textbf {x}^{\ast },\boldsymbol {\xi })]-\gamma \rho ^{-}[\pi _{i}(x_{i}^{\ast },\xi _{i})]\) to model (11) is shown in Table 9.
Table 9

The calculation results of the maximum objective function value

Maximum objective

Risk measure parameter

function value($)

γ = 0

γ = 0.5

γ = 1

γ = 1.5

γ = 2

γ = 2.5

e

21971.09

21393.51

20968.18

20618.12

20300.69

20004.55

From Table 9 and Fig. 3, we can see the greater the risk measure parameter γ, the higher the inventory profit E[π(x, \(\boldsymbol {\xi })]-\gamma \rho ^{-}[\pi _{i}(x_{i}^{\ast },\xi _{i})]\). As a consequence, the computational results demonstrate that the absolute semi-deviation risk criterion in the current development is a new approach to modeling multi-item inventory problem, it can provide diversification order policies for this inventory problem.
Fig. 3

The relationship between the expected profit with risk and risk measure parameter γ

The results of models (9)–(11) revel a fact that the profit and risk are mutually restricted. If we want to obtain a higher profit, it is bound to bring a greater risk. However, such a conclusion for our models can be drawn through observing Figs. 12 and 3. There are critical points make both profit and risk will not increase without limit.

6 Conclusions

This paper studied the multi-item single-period inventory problem with fuzzy demand, and proposed a novel absolute semi-deviation risk measure for gauging the risk. The major conclusions include the following several aspects:
  1. (i)

    We built three multi-item single-period inventory profit models, in which the uncertain demands were described by possibility distributions. In order to solve the models we built, we adopted the min-max modeling method to obtain the equivalent problem of the original problem. Since the expression of the optimal order policy contained E[(1/ξi −E[1/ξi])+] (i = 1,2,…, n), we calculated the absolute semi-deviation value about the reciprocal of the demand.

     
  2. (ii)

    We addressed the cases of fuzzy demands followed triangular, trapezoidal and Erlang possibility distributions. The computational results had been summarized in Theorems 1–3, which could help us to obtain the analytic solution or general solution to the proposed equivalent models.

     
  3. (iii)

    In accordance with the obtained theoretical results, a numerical experiment was conducted to illustrate the proposed methods and three examples were presented. The obtained optimal order policies were reported in Tables 46 and 8 respectively. What’s more, the computational results supported our arguments.

     

For the sake of computing tractability of the models, some simplifications were taken and only tree common fuzzy distributions were listed. Nevertheless, limitations can be relaxed in future studies. For the future work, the presented models in this paper can still be worked on under the relaxed conditions. Furthermore, we will work with the inventory problem and extend the problem to multi-period situation.

Notes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.61773150).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHebei UniversityBaodingChina
  2. 2.School of Economics and ManagementBeijing University of Chemical TechnologyBeijingChina

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