A riskaverse multiitem inventory problem with uncertain demand
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Abstract
The observed values of demands in reallife 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 riskaverse fuzzy optimization method is adopted for the multiitem 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 semideviation with expected value operator and then model analysis is given for the minmax inventory models. To make the inventory problem tractable and computable, the equivalent forms of the proposed optimization models are discussed. Subsequently, several useful absolute semideviation 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 semideviation Fuzzy optimization1 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 valueatrisk (VaR), conditional valueatrisk (CVaR) and semideviation. 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.
Absolute semideviation, as a kind of downside risk measure, is employed in the multiitem singleperiod 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 semideviation 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 riskaverse biobjective model and three equivalent single objective models for the multiitem singleperiod inventory problem and then presents the model analysis. Section 4 discusses the equivalent problem with mutually independent demand under absolute semideviation risk criterion, and deals with the computing of the absolute semideviation 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 lowcost 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 endproduct 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 safetystock determination approach under stochastic demand and random supply yield. AmiriAref et al. (2018) dealt with a multiperiod locationinventory optimization problem in a multiechelon supply chain network characterized by an uncertain demand and a multisourcing feature.
The observed values of demand and supply in realworld 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 multiitem 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 twoechelon 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 singleorder 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 twostage optimization method for multiobjective supply chain network design (MOSCND) problem with uncertain transportation costs and uncertain customer demands, and developed two objectives for the SCND problem on the basis of riskneutral and riskaverse criteria. A fuzzystochastic 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 singleperiod inventory problem, in which the uncertain market demand is characterized by generalized parametric intervalvalued 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 riskaverse single period inventory problem. Shi et al. (2019) presented a timeconsistent 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 meanvariance 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 multiperiod risk minimization inventory models for fashion product purchasing via a meanvariance approach. By the expected utility functions method, Choi and Park (2015) studied a few dynamic riskaverse inventory models by using additive utility functions. There are also investigators taking semideviation 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 riskneutral manufacturer and a riskaverse 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. Semideviation 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 semideviation, 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 semideviation 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 semideviation.
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 semideviation risk measure (one method that gauging the deviation levels from the expected profits), we develop three novel inventory models to optimize the multiitem inventory problem for the risk averse decision maker in a fuzzy environment. The credibilistic optimization approach is employed to construct the minmax inventory models and the fuzzy possibility distributions are adopted to describe the uncertain demands.
3 Multiitem singleperiod 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 multiitem singleperiod inventory problem under riskaverse 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 multiitem 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 = [c_{1}, c_{2},⋯ , c_{n}]:

Unit fixed costs per inventoried item;
 d = [d_{1}, d_{2},⋯ , d_{n}]:

Unit revenues per inventoried item;
 g = [g_{1}, g_{2},⋯ , g_{n}]:

Unit holding costs per inventoried item;
 ξ = [ξ_{1}, ξ_{2},⋯ , ξ_{n}]:

Fuzzy demand vector in the inventory problem;
 x = [x_{1}, x_{2},⋯ , x_{n}]:

Order quantity vector in the inventory problem;
 γ:

Absolute semideviation parameter;
 n:

Numbers of the inventoried item.
If decision makers want to obtain the maximum profit under riskaverse criterion, they usually use the expected value and riskaverse 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 semideviation criterion
In this subsection, we mainly introduce the absolute semideviation risk criterion and discuss the formulation of models for the multiitem inventory problem.
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 semideviation as a kind of downside risk measure occurs in the inventory problem, and represents the risk aversion attitude of decision makers. Under the riskaverse criterion, we take the absolute semideviation (Chen et al. 2012) as a risk measure that the decision maker can maximize the profit and minimize the risk at the same time.
For the absolute semideviation criterion denotes the degree of deviation from the risk in the inventory problem, then we take the maximum of the absolute semideviation for the i th profit function as an objective function, and it reads that
3.3 Model analysis
According to the minmax modeling method, we transform models (3)–(5) into their equivalent forms, which are as follows.
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:
Through analyzing the models for the multiitem inventory problem, we have gotten the equivalent models by the minmax modeling method. In next section, we will discuss the specific equivalent forms of models (6)–(8) when the demands ξ_{i}, i = 1,2,…, n are mutually independent fuzzy variables.
4 Equivalent problem under absolute semideviation criterion
4.1 Forms of absolute semideviation under independence
In order to solve models (6)–(8), the key point is to compute the expected value and absolute semideviation of i th profit function π_{i}(x_{i}, ξ_{i}). In the section, we will discuss the calculation of E[π(x,ξ)] and ρ^{−}[π_{i}(x_{i}, ξ_{i})].
 (i)if b_{i} ≥ 0, then the equivalent expression is$$ \mathrm{E}\left[b_{i}\left( \frac{1}{\xi_{i}}\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)\right]^{}=b_{i}\mathrm{E}\left( \frac{1}{\xi_{i}}\mathrm{E}\left[\frac{1}{\xi_{i}}\right]\right)^{}, $$
 (ii)if b_{i} < 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.
In order to solve these models,we need to compute the values of E[1/ξ_{i}] 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/ξ_{i}] in Li and Liu (2016) when demand ξ_{i}, i = 1,2,…, n obey common fuzzy distributions. In the next section, we will discuss the equivalent forms of absolute semideviation 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 semideviation
In this section, we will deal with the computing of the absolute semideviation 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 (r_{1}, r_{2}, r_{3}) withr_{1} > 0 andm = E[1/ξ],then the absolute semideviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }m\right )^{+}\right ]\)iscomputed by
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 (r_{1}, r_{2}, r_{3}, r_{4}) withr_{1} > 0 andm = E[1/ξ],then the absolute semideviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }m\right )^{+}\right ]\)iscomputed by
The proof of Theorem 2 is shown in Appendix.
Finally, for the Erlang fuzzy demand distribution, we have related results as follows:
Theorem 3
 (i)
If\(\frac {1}{r_{2}}\leq m<\frac {1}{\lambda r}\)andr = 1,then the absolute semideviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }m\right )^{+}\right ]\)iscomputed by
where the expected valuem is computed 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)$$ m=\frac{e}{2\lambda}\left[\ln\frac{\lambda^{2}}{r_{1}r_{2}}+\sum\limits_{i=1}^{+\infty}\frac{2(1)^{i}\left( \frac{r_{2}}{\lambda}\right)^{i}\left( \frac{r_{1}}{\lambda}\right)^{i}}{i\cdot i!}\right]+\frac{1}{\lambda}\frac{1}{r_{2}}. $$(14)  (ii)If\(\frac {1}{r_{2}}\leq m<\frac {1}{\lambda r}\)and r = n,(n > 1), then the absolute semideviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }m\right )^{+}\right ]\)is computed by$$ \begin{array}{@{}rcl@{}} \mathrm{E}[(\etam)^{+}]&=&\left( \frac{1}{r_{1}}m\right)\left[1\frac{1}{2}\left( \frac{r_{1}}{\lambda r}\right)^{r}e^{r\frac{r_{1}}{\lambda}}\right]\left( \frac{1}{r_{1}}\frac{1}{\lambda r}\right)+\frac{e^{r}}{2\lambda}\left( \frac{1}{r}\right)^{r}\left[\left( \frac{1}{\lambda m}\right)^{r2}e^{\frac{1}{\lambda m}}\right.\\ && + \left. \left( \frac{r_{1}}{\lambda}\right)^{r2}e^{\frac{r_{1}}{\lambda}}2r^{r2}e^{r}+(r2)\left( {\int}_{(\frac{1}{\lambda m},r)}t^{r3}e^{t}\text{dt}{\int}_{(r,\frac{r_{1}}{\lambda})}t^{r3}e^{t}\text{dt}\right)\right], \end{array} $$(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)  (iii)If\(\frac {1}{\lambda r}\leq m \leq \frac {1}{r_{1}}\)and r = 1, then the absolute semideviation 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).
 (iv)
If\(\frac {1}{\lambda r}\leq m \leq \frac {1}{r_{1}}\)and r = n,(n > 1), then the absolute semideviation value\(\mathrm {E}\left [\left (\frac {1}{\xi }m\right )^{+}\right ]\)is computed by
where the expected value m of η is computed by the formulation (16).$$ \begin{array}{@{}rcl@{}} \mathrm{E}[(\eta  m)^{+}]&=&  \frac{1}{2}\left( \frac{1}{r_{1}}  m \right) \left( \frac{r_{1}}{\lambda r}\right) ^{r} e^{r  \frac{r_{1}}{\lambda}}\\&&+ \frac{e^{r}}{2\lambda} \left( \frac{1}{r}\right)^{r} \left[ \left( \frac{r_{1}}{\lambda}\right)^{r  2}e^{  \frac{r_{1}}{\lambda}}  \left( \frac{1}{\lambda m}\right) ^{r  2}e^{  \frac{1}{\lambda m}}\right.\\ &&  \left. (r2) {\int}_{\left( \frac{1}{\lambda m},\frac{r_{1}}{\lambda}\right)} t^{r  3}e^{  t}\text{dt} \right], \end{array} $$(18)Note that when r_{1} = 0, the absolute semideviation 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
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)
The distributions of fuzzy demands ξ_{i}
Item  Fuzzy demands ξ_{i}  Item  Fuzzy demands ξ_{i}  Item  Fuzzy demands ξ_{i} 

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) 
The calculation results of the expected value and absolute semideviation
Expected value and  Item  

absolute semideviation  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 semideviation  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 
Optimal order policy under different minimum acceptable profit levels
Minimum acceptable  Optimal order policy  

profit level  x_{1}  x_{2}  x_{3}  x_{4}  x_{5}  x_{6}  x_{7}  x_{8}  x_{9}  x_{10} 
r_{0} = 5000  59  90  37  63  51  48  42  19  20  24 
r_{0} = 10000  129  193  81  138  110  104  91  42  45  53 
r_{0} = 15000  218  328  138  234  187  177  155  71  76  91 
r_{0} = 20000  331  530  239  416  334  285  276  126  135  161 
r_{0} = 21970  331  528  239  500  505  290  390  194  254  345 
The calculation results of the minimum objective function value
Minimum objective  Minimum acceptable profit level  

function value($)  r_{0} = 5000  r_{0} = 10000  r_{0} = 15000  r_{0} = 20000  r_{0} = 21700 
\(\max _{1\leq i\leq n}\rho ^{}[\pi (x_{i}^{\ast },\xi _{i})]\)  7.29  33.86  97.37  308.21  1398.54 
Computational results of model (10)
The possibility distributions of demands ξ_{i},(i = 1,2,…,10) are provided in Table 2. The expected value E[1/ξ_{i}] and the absolute semideviation value E[(1/ξ_{i} −E[1/ξ_{i}])^{+}], i = 1,2,…,10 are shown in Table 3.
Optimal order policy under different maximum acceptable risk levels
Maximum acceptable  Optimal order policy  

risk level  x_{1}  x_{2}  x_{3}  x_{4}  x_{5}  x_{6}  x_{7}  x_{8}  x_{9}  x_{10} 
s_{0} = 50  156  235  99  167  134  126  111  51  54  65 
s_{0} = 100  221  333  140  237  190  179  157  72  77  92 
s_{0} = 500  331  528  239  500  425  289  351  161  173  206 
s_{0} = 1000  331  528  239  500  505  289  391  195  244  291 
s_{0} = 1500  331  528  239  500  505  289  391  195  254  352 
The calculation results of the maximum objective function value
Maximum objective  Maximum acceptable risk level  

function value($)  s_{0} = 50  s_{0} = 100  s_{0} = 500  s_{0} = 1000  s_{0} = 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 s_{0} is varied, the inventory total expected profit E[π(x^{∗},ξ)] is varied accordingly. The computational results demonstrate that the greater the riskseeking, the higher the profits can be obtained. It is also meaningful in the real life.
Computational results of model (11)
Optimal order policy under different risk measure parameters
Risk measure  Optimal order policy  

parameter  x_{1}  x_{2}  x_{3}  x_{4}  x_{5}  x_{6}  x_{7}  x_{8}  x_{9}  x_{10} 
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 
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 
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. 1, 2 and 3. There are critical points make both profit and risk will not increase without limit.
6 Conclusions
 (i)
We built three multiitem singleperiod inventory profit models, in which the uncertain demands were described by possibility distributions. In order to solve the models we built, we adopted the minmax modeling method to obtain the equivalent problem of the original problem. Since the expression of the optimal order policy contained E[(1/ξ_{i} −E[1/ξ_{i}])^{+}] (i = 1,2,…, n), we calculated the absolute semideviation value about the reciprocal of the demand.
 (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.
 (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 4, 6 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 multiperiod situation.
Notes
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No.61773150).
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