Optimal Policies of Newsvendor Model Under Inventory-Dependent Demand Ting GAO * and Tao-feng YE

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1 207 2 nd International Conference on Education, Management and Systems Engineering (EMSE 207 ISBN: Optimal Policies of Newsvendor Model Under Inventory-Dependent Demand Ting GO * and Tao-feng YE NO. 2, Mengxi Road, Jingkou District, Zhenjiang City, Jiangsu Prov. China *Corresponding author Keywords: Inventory-dependent demand, CVaR criterion, Risk-averse, Newsvendor. bstract. This paper aims to give the newsvendor s unique optimal order quantity in two forms of models under inventory-dependent demand: the linear model and the non-linear model (e.g. exponential function, and study differences caused by different effects of the inventory level on the consumer demand. Finally, we provide optimal order policies and find the risk-averse newsvendor s optimal order quantity always decreases in the risk aversion and is still lower than that of the risk-neutral newsvendor no matter how the inventory level affects the consumer demand. Moreover, effects of parameters on optimal order quantities have been proved through numerical studies. Introduction t present, in a large number of shopping malls, in order to attract consumers, businesses trend to fill with more goods in a limited space (such as promotion display of milk or fruit, etc.. This phenomenon is called inventory-dependent demand. Literatures related with inventory-dependent demand can be summarized into two categories, forms in which the demand rate of an item was a function of the initial inventory level and those in which was dependent on the instantaneous inventory level[]. Chang et al put maximize profit as the objective function, under the relationship of instantaneous inventory-dependent demand, to provide existential condition of optional solution and a series of algorithms for solving[2]. Wang and Vakharia proposed that market demand rely on the initial inventory level of the linear demand, and then compared with the traditional newsvendor model[3]. HN Soni extended the inventory model proposed by Chang et al from two aspects: The demand rate as multivariate function of price and level of inventory; Delay in pay mentis permissible[4]. Chen et al established provided periodic-review inventory systems with inventory-dependent demand, including multiplicative demand model and additive demand model and provided optimal replenishment policies[5]. Saha and Goyal, through applying bargaining theory, had established that stock elasticity played an important role to select coordination contract and also determined a threshold value stock elasticity[6]. Parthasarathi et al assumed that demand was a linear function of the stock level and the selling price, and finally concluded that quantity discounts and return policies played important roles in the supply-chain coordination[7]. Panda also concluded that conventional revenue sharing contract could not coordinate the system but revenue and cost sharing (RCS contract was able to coordinate the system and led to a win win outcome under stock-price dependent demand[8]. ll of these literatures assumed the decision makers were the risk-neutral newsvendors. However, newsvendors generally are regarded as risk-averse decision makers. Xu and Li investigated a risk-averse newsvendor model by balancing the expected profit and CVaR criterion in a newsvendor model setting[9]. Choi et al considered a multiproduct risk-averse newsvendor under the law-invariant coherent measures of risk, and showed that increased risk aversion leads to decreased orders[0]. Wu et al showed that capacity uncertainty decreased the order quantity under the CVaR criterion considering the risk-averse newsvendor model with random capacity[]. Comparing with different methods of measuring risk, Katariya analyzed the optimal order quantity of the risk-neutral newsvendor model and that of risk-averse newsvendor model[2]. These papers all explained that the risk aversion had a great on decision makers. 49

2 bove referenced literatures either discussed the inventory-dependent demand, or focused on the risk-averse newsvendor. Our aims are to find the necessary and sufficient conditions of the existence and uniqueness of the risk-averse newsvendor s the optimal solution with inventory-dependent demand when demand is a linear function of stock level or a non-linear function. Model Basics Parameters This paper considers only a one period setting for a risk-averse newsvendor retailer facing inventory-dependent demand and assumes that the retailer can obtain completely the information in the market. The retail price per unit charged by the retailer is p, c is the manufacturing cost per unit, and r is net salvage value per unsold unit. In addition, in order to ensure the practicality of the parameters, the following assumption that p > c > r > 0is made in this paper. Performance Measure: CVaR In this paper, we will research risk-averse newsvendor s optimal order quantity and compare with that of the risk-neutral newsvendor. CVaR criterion focuses on maximizing the average profit falling below a certain quantile level that is defined as the maximum profit at a specified confidence level. Since the CVaR measure takes both reward and risk into account, it is extensively used in the literature, see, e.g. Chen, Y., & Zhang, Z. G.[3], Wu et al[4]. In our context, the newsvendor s objective under the CVaR measure can be represented as we adopt Conditional Value-at-Risk method (CVaR to further study risk-averse newsvendor s optimal order. Then, the objective function under the CVaR method is as follows: CVaR ( π( = max g(, ϕ = ϕ E[ ϕ π ( ] ϕ R where, ϕ is the target profit of the risk-averse decision maker, (0,] reflects the degree of risk aversion of the newsvendor, the smaller the value of indicates that the risk-averse newsvendor has a greater degree of aversion to risk, represents the risk-averse newsvendor s order quantity at the beginning of the selling period, π ( and E ( stands for the newsvendor s profit and expected profit respectively. Newsvendor s Optimal Policy The Linear Model In this part, the linear demand function of stock level is assumed as follows (see e.g. Devangan[5]: D = α β ε (2 where, D is the market demand which the retailer faces, is the retailer's order quantity, α > 0 is the initial demand when the retailer's demand isn t affected by the inventory level, β is the stock dependent factor ( 0 < β <,In real life, it is commonly accepted that per additional product ordered by the retailer can cause the increase of demand, but it is impossible to reach one.ε is a random variable of demand which caused by exogenous variables ( ε [, B], 0 < < B and E( ε = µ. fi ( is the probability density function of random variable ε, and Fi ( is the cumulative distribution function of random variable ε. So, in this linear model, we have known that the newsvendor s profit π is expressed by ( p c ( p r( D, which is a function of and combines with ( and (2, we can have ϕ ( c r g(, ϕ = ϕ df( ε ϕ ( p c df( ε ( p r( α β ε ( α β B [ ] (3 ( α β Then, we can obtain the risk-averse newsvendor s optimal order quantity under the CVaR criterion as the following proposition. 50 (

3 Proposition If the decision maker is a risk-averse newsvendor, the unique optimal order quantity is expressed as: * ( p c CVaR = α F [ ] ( β ( β ( p r From Proposition, we can know the risk-averse newsvendor s optimal order quantity is increasing inα, p and β, but decreases in c, because the consumer demand increases inα, and β, which lead to increase the risk-averse newsvendor s optimal order quantity. Meanwhile, it is evident to find that it is decreasing in, which means the risk-averse newsvendor orders less than risk-neutral newsvendor s optimal order quantity, and it is equal that where =. Meanwhile, analyzing the linear demand function and assuming β =0, which means consumer demand is independent on stock level, it can obtain the risk-averse newsvendor s optimal order quantity was * ( p c CVaR U = α F [ ], which ( p r * * compared with Proposition, we can find CVaR > CVaR U and it results from stock factor has an equally galvanizing impact on consumer demand, which causes the risk-averse newsvendor s to increase the optimal order quantity. Finally, we can conclude that the risk-averse newsvendor s optimal order quantity decrease in the risk aversion and the impact of stock level on demand leads risk-averse newsvendors to order more than the risk-averse newsvendor s optimal order quantity under the condition stock level has no impact on demand. The Non-linear Model The former part, we had studied related problems under the linear model, through plenty of literatures have stated that the improvement of inventory level, which has a positive impact on consumer demand, there are a lot of the different concrete function forms. It can be generally divided into two categories: linear model and non-linear model. Now we have analyzed the former. In this part, we will combine the exponential function models to further clarify the non-linear model with the inventory-dependent demand. Therefore, the exponential function of consumer demand is expressed as: D = ε ( > 0, 0 < < (4 here, is the scale coefficient, and is the stock dependent factor, which equals to β and meets 0 < <. We assume that other parameters remain unchanged. In this case, according to the expected profit maximization and (4, we can derive the following proposition. Proposition 2 If the demand is a non-linear function of the inventory-dependent, the risk-neutral newsvendor s optimal order quantity satisfies the following equation: ( E ( E ( E p c F( d ( F( ε ε = p r Proof We had know the newsvendor s profit is ( p c ( p r( D demand function. It can obtain, and (4 provides the [ ] E π ( = E ( p c ( p r( ε = ( p c ( p r F( ε dε Then, we can have the first derivative of that de [ π ( ] = ( p c ( p r F ( ε dε ( p r ( F ( d (5 By making (5 be zero, we can obtain the risk-neural newsvendor s optimal order quantity satisfies 5

4 ( ( E E p c the condition that F( ε dε ( F( ( =, and the proposition can be proved. p r E We can see the optimal order quantity under the non-linear model of inventory-dependent demand is different from that under the linear model, which shows the form of demand function of inventory level can also have a great influence on the newsvendor s optimal order quantity. We further study the risk-averse newsvendor s optimal order quantity based on ( and (4, it has g(, ϕ in the CVaR criterion as: ϕ ( r c B [ ] (6 g(, ϕ = ϕ df( ε ϕ ( p c df( ε ( p r ε Similarly, we can come to the following proposition about the risk-averse newsvendor s optimal order quantity. Proposition 3 Under inventory-dependent demand, a risk-averse newsvendor s optimal order quantity satisfies the following equations: CVaR ( ( CVaR ( p c F ε dε F = ( p r ( CVaR ( ( (. Proof We further analyze (6 Case, ϕ ( r c, we have g(, ϕ = ϕ, and g(, ϕ / ϕ =. ϕ ( r c ( Case2, ( r c < ϕ ( p c, (, p r g ϕ = ϕ ϕ ( r c ( p r ε df( ε g(, ϕ ϕ ( r c = F( ( p r ϕ where, we can obtain two specific values that and and g(, ϕ ϕ ϕ = ( r c = g(, ϕ and ϕ = ( pc = F( ϕ B Case 3, ϕ > ( p c, g(, ϕ = ϕ ϕ ( r c ( p r ε df( ε [ ϕ ( p c ] df( ε g(, ϕ / ϕ = < 0., So, let ϕ ( be the optimal solution, combining Cases -3, we obtain ϕ ( ( ( r c,( p c ]. nd it is limited by the value of F(. Now, if ( ( < F, then ( ( ϕ = p c. g(, ϕ ( = ( p c ( p r F( ε dε Then, we can derive the first derivative and the second derivative as follows ( p c ( p r F( ε dε ( p r ( F( dg(, ϕ ( = d ( 2 dg (, ϕ ( 2 ( p r nd 2 ( ( = p r F ( ε dε F ( 0 d < Therefore, we can know g(, ϕ ( is a concave function of, and let dg(, ϕ ( / d be zero, we can have the risk-averse newsvendor s optimal order quantity meet the following equation 52

5 ( p c ( ( p r F( ε dε ( F( =. While > F (, then ϕ ( ( r c,( p c ] and ϕ ( = ( p r F ( ( r c. So we have ( F g(, ϕ = ( p r F ( ( r c ( p r F ( ε df ( ε dg(, ϕ F Thus ( = ( r c ( p r ε df( ε 0 d >, there is no optimal solution. So, this proposition is proved. From proposition 2 and proposition 3, we also can derive another proposition. Proposition 4 In the non-linear model with inventory-dependent demand, the risk-averse newsvendor s optimal order quantity decreases in the risk aversion and is lower than that of the risk-neutral newsvendor. Specially, if =, it is equal to that of the risk-neutral newsvendor s optimal quantity. τ Proof We assume that ( F ( d ( F ( Γ τ = ε ε ( τ τ, where τ = /. Then, there τ dγ( τ ( is ε df ε = ( f ( τ, and it can be known that τ is a monotonically increasing function dτ τ of. Therefore, we obtain dγ ( τ / dτ > 0, which means that Γ( τ is a monotonically increasing function of. Now, it is evident to see CVaR decreases in the risk aversion because Γ( τ increases in, which shows CVaR < E ; and if =, CVaR = E. ccording to above proposition, the relationship between the risk-averse newsvendor s optimal order quantity and that of the risk-neutral newsvendor in the non-linear model is shown. Furthermore, based on the proof the proposition 4, we can also find the risk-averse newsvendor s optimal order quantity decrease in p and r, but decreases in c, which can be proved when the retailer is a risk-neutral newsvendor. That means the relationship between the risk-averse newsvendor s optimal order quantity and that of the risk-neutral newsvendor is not influenced by the functional forms of the inventory-dependent demand and the impact of parameters have no change. Besides, analyzing the equation that the risk-neutral newsvendor s optimal order quantity satisfies, ( we can find τ = / is fixed when the selling price, the manufacturing cost and the net salvage value have no change, i.e. ( p c / ( p r is fixed, meanwhile, it is not difficult to understand that the risk-neutral optimal order quantity is decreasing when increases, but increases in, which can be proved to be true when the retailer is the risk-averse newsvendor. Numerical Studies Table. Risk-averse optimal order quantities. β This part will analyze further effects of stock level and the risk aversion by assuming random variable 53

6 of demand is uniformly distributed over [ 0,0 ], which indicates µ =5. nd other parameters are defined as: p = 0, c = 6 and r = 2. In the linear model, we assume D = 0 β ε. By means of MTLB, we have risk-averse newsvendor s optimal order quantities adjusting different β and to find effects of stock level and risk aversion as Table. From table. we can know the risk-averse newsvendor s optimal order quantity decreases in the risk aversion but increases in the stock level, which is associated with those analyzed from proposition. So, for certain good whose demand is a linear function of stock level, this optimal order policy is practical for retailers to make decisions. In order to figure out them in the non-linear model, similarly, we turn to MTLB and demand function is expressed as D = 0ε, and others parameters are same to those in linear model. So, optimal order quantities in non-linear model are shown in Table 2. It also shows the risk aversion has a passive impact on the risk-averse newsvendor s optimal order quantity, while the optimal order quantity increase in the stock level, which all are same to those in the linear model. However, from table 2, we can see that the optimal order quantity with the inventory level has increased dramatically, which far surpassed it in the linear model. It means that the effect of stock level in form of non-linear function is much greater than that in form of linear function. Besides, existing literatures had explained that stock coefficient in the demand function is often not too big; it, this category of goods are considered to be influenced by stock level sp easily,, which can be explained. is that if β > 0.4 or > 0.4 so these order quantities in table 2 are too great if β > 0.4 Table 2. Optimal order quantities in the non-linear model Conclusions We had studied two newsvendor models under inventory-dependent demand, including the linear model and the non-linear model. We propose newsvendor s optimal policies under different conditions and find the risk-averse newsvendor s optimal order quantity decreases in the risk aversion and is always lower than that the risk-neutral newsvendor under inventory-dependent demand. Further, it is greater than the risk-averse newsvendor s optimal order quantity in the classical model. Meanwhile, effects of parameters on optimal order quantities are shown and the influence of the inventory level on the consumer demand cannot cause changes of the relationship between the risk-averse newsvendor s optimal order quantity and that of the risk-neutral no matter what forms that stock level affects demand in. References [] Urban T L. Inventory models with inventory-level-dependent demand: comprehensive review and unifying theory [J]. European Journal of Operational Research, 2005, 62(3: [2] Chang C T, Teng J T, Goyal S K. Optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand [J]. International Journal of Production Economics, 200, 23(:

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