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1. Introduction

 In a classic newsboy problem the newsboy should decide how many newspapers to order for his daily business very early in the morning. Not only too many newspapers but also too few will incur him unnecessarily high costs. In other words, in case he orders too many he will struggle with left-over newspapers at the end of day, while he would have missed an opportunity for incremental profit in case he orders too few in the morning. The most important reason why this classical newsboy problem has been most popular research topic for decades is that it allows researchers to investigate a variety of single period inventory problems under so many different conditions and scenarios such as the max(or target) capacity constraint, target production levels, target profit lev-els, various pricing schemes, underlining detail inventory holding/ shortage cost structures and so on. The importance of the newsboy research problem can also be easily proved by counting the number of OM/OR research articles pub-lished so far. The number of articles is close to 400, which use the words ‘newsboy’ or ‘newsvendor’.

 Similar to the structure of inventory problems the newsboy problem can have following structures where daily demand and daily supply can be either deterministic or stochastic.

 In the original newsboy problem setting discussed in Morse and Kimball [12] stochastic demand with determin-istic supply situation has been assumed. Morse and Kim-ball’s [12] research was followed by numerous researchers such as Hanssman [8], Porteus [13], Schwerizer and Cachon [15] to deal with more sophiscated and realistic problems while maintaining the basic framework of stochastic demand and deterministic supply.

 Even though stochastic demand assumption with determin-istic supply is adopted in most of the single-period inventory model, in many real-life situations, one can easily observe random gaps between the originally placed order quantity and the actually achieved quantity. Increased popularity of global sourcing is one big reason for generating less than perfect supply processes from suppliers to retailers. For ex-ample, to reduce purchase costs and attract a larger base of customers, retailers such as Wal-Mart, Home Depot and Dollar General are constantly seeking suppliers with lower prices and finding them at greater and greater distances from their distribution centers (DCs) and stores. Consequently, a significant proportion of shipped products from overseas sup-pliers is susceptible to defects. Reasons for defects include missing parts, misplaced products (at DCs, stores) or mis-takes in orders and shipments. A similar example could be a typical production line where the production yield assumes less than 100% resulting in a different number of goods man-ufactured than originally planned. In these situations, the problem is how to choose the size of an order or how many parts to begin production to meet one time fixed demand. See Kim et al. [9] for precise problem formulation and solution analytics.

 The basic newsboy research framework produced some oth-er meaningful research topics incorporating uncertain supply situations. Newsboy problem research with uncertain supply framework was initiated by Silver [17] and followed by many other researchers. Silver [17] is one of the earliest pa-pers on the uncertain supply under economic order quantity (EOQ) framework. He studied two cases, in the first case, the standard deviation of the amount received is independent of the lot size, while in second case the standard deviation is proportional to the lot size. One of the interesting results among his findings was that the optimal order quantity depends only on the mean and the standard deviation of the amount received. Yano and Lee [19] is the most popular reference.

 Newsboy problems so far have not considered the news-boy’s attitude towards various potential risks such as finan-cial risk, meeting the target profit or cost etc. In other words researchers have assumed the newsboy’s indifference on those risks and focused on developing risk neutral optimal sol-utions optimizing the expected profit or cost.

 To overcome this ‘flaw of average’ in solving the sin-gle-period inventory problem many researchers studied the behavior of Risk Averse Newsboy. These include Spulber  [18], Bouakiz and Sobel [2], Eeckhoudt et al. [5], Agrawal and Seshadri [1], Chen and Federgruen [3], Seifert et al. [16], Chen et al. [4], Haksoz and Seshadri [7]. Lau [11], Gan et al. [6] examined newsvendor solutions which maximize expected utility. Gan et al. [6] also investigated the new ob-jective function of maximizing the probability of achieving a budgeted profit. Eeckhoudt et al. [5] examined the risk and risk aversion in a single-period inventory problem where demand is stochastic while supply is deterministic. They show that the optimal order quantity decreases as decision maker’s risk-aversion increases because a lower order amount definitely reduces the inherent risks of the outcome. In Bouakiz and Sobel [2], they explored the newsvendor problem with the exponential utility and showed that a base- stock policy is optimal when a multi-period newsvendor problem is opti-mized with an exponential utility criterion. Agrawal and Seshadri [1] also investigated the newsvendor problem with the objective being maximizing the expected utility. In their problem setting, both price and order quantity are decision variables for the risk-averse retailer.

 In this paper, the research output of Eeckhoudt et al. [5] is revisited to the case of uncertain supply situation. In Eeck-houdt et al. [5] they started with the basic newsboy problem, where the demand is stochastic while supply is deterministic, then deployed the utility functions over the newsboy’s ex-pected profit function to embed his risk attitude into the problem. From their research they derived and summarized comparative statics of the risk-averse newsboy as the changes of the optimal order quantity. To our best knowledge and literature review results, no article on the risk-averse news-boy problem under uncertain supply problem has been published.

 The rest of this paper is organized as following: Section 2 reminds of the basic newsboy problem framework with uncer-tain supply setting and the risk neutral optimal order quantity as well. In Section 3, we introduce the risk-averse newsboy problem framework followed by a derivation of characteristics of the optimal order quantity. In Section 4, we present a brief numerical study to demonstrate the result from Section 3. Finally, in Section 5, we conclude this paper by summarizing the findings and insights throughout our journey.

2. Previous Model:Risk-Neutral News-boy Problem with Uncertain Supply

 In this section we consider a newsboy problem with uncer-tain supply instead of uncertain demand in the classical news-boy problem. And we assume θ, the variable representing daily demand, is fixed and known. This assumption might be unrealistic but in our search the main purpose is to verify the impact of risk-aversion on the optimal order quantity un-der unreliable supply condition. To focus on the research purpose we sacrifice the uncertain demand assumption. Under this assumption, when the newsboy’s order quantity is α, the number of arrived newspapers to the newsboy is Y α, where Y represents random yield proportion of α with distribution function G(y) (p.d.f. g(y)). Let’s define p as retail price and c as wholesale price. And all unsold news-papers are returned to the distributor office at salvage price v. Finally the newsboy is allowed to obtain additional news-papers if demand is greater than what he has on hand, but at a higher cost, c^. As addressed in Eeckhoudt et al. [5] a natural assumption is that 0 ≤ v＜ c＜ c^ ≤ p Then, the function, Z(Y, α), represents the newsboy’s total revenue at the end of each day. The newsboy, facing an uncertain supply via random yield process, has to determine α, the size of his original newspaper order early in the morning.

 or equivalently, using the two mutually exclusive ranges, θ ≤ Yα, θ ＞Yα, Z(Y, α)  can be rewritten as following

 Z₁(Y, α) = (p-v)θ-(c-v)Yα
 if θ ≤ Yα,
 Z₁(Y, α) = (p+c^-c)Yα-c^θ.

 otherwise.

 Using the above equations, the expected revenue function can be expressed as:

 where E[⋅] denotes the expectation operator.

 The expected revenue function is concave in the order quantity (or batch size), α. Readers can refer to Kim et al. [10] for rigorous proof of concavity. And the concavity of the expected revenue function allows us to rely on the first order condition to find the optimal batch size which max-imizes the expected revenue. From the simplification of the first order condition, it can be shown that the risk-neutral solution α^* should satisfy the following equation:

 For interested readers the derivation of above expression (3) can be found in Kim et al. [10].

 Once the distribution of Y and the demand level, θ are specified, the corresponding solution can be computed using the above equation.

3. Risk-Averse Newsboy Problem with Uncertain Supply

 Eeckhoudt et al. [5] showed that the risk-averse newsboy always would order fewer newspapers and they proved that this quantity would decrease as the newsboy’s risk-aversion increases. But, under random supply situation, this is not quite true and we will discuss about it in this section.

 Similar to our research, Kim et al. [10] considered the impact of the risk aversion on the optimal order quantity under supply uncertainty. They introduced downside-risk constraint to reflect the newsboy’s risk attitude on his expected revenue at the end of a business day. The downside-risk constraint enables the newsboy to constrain the probability of meeting his target revenue to a desirable level. Readers can think of the following form of downside-risk constraint for intuitive understanding : P(Z(Y, α) ≤ τ₁) ≤ w₁.

 In addition, the downside-risk parameter pair(τ₁, w₁) ex-hibits higher risk aversion than another pair(τ₂, w₂) when-ever (τ₁ ≥ τ₂) and (w₁ ≤ w₂). Suppose that w₁ and w₂ are same at 95% level and τ₁= 100, τ₂= 75, respectively. Then the corresponding downside-risk constraint with (τ₁, w₁) im-plies that the probability of the newsboy’s payoff being less than or equal to 100 should be no greater than 95% while constraint with (τ₂, w₂) implies the payoff being no greater than 75 should be 95% or smaller. In this way, whenever τ₁ ≥ τ₂ downside-risk constraint with the parameter with τ₁ represents more risk-averse attitude of the newsboy than that with the parameter τ₂. In their article, Kim et al. [10] pre-sented numerical examples to show that as the risk aversion increases the size of newspaper order placed by the newsboy also increases when the supply probability is not certain. In other words, the more the newsboy exhibits risk aversion, the more he orders the newspaper to reflect his risk attitudes. But in this paper we show that for the same problem setting the more the newsboy exhibits risk aversion does not guarantee the increase in the newspaper order size. We adopted concave transformation of the utility function to see the effect of news-boy’s increased risk-aversion in our search rather than adopting the downside-risk aversion constraint as in Kim et al. [10].

 In our paper, similar to Eeckhoudt et at. [5], the newsboy’s preference over the final wealth is assumed to be of the ex-pected-utility type where u(⋅) is the corresponding utility function. Then, the risk-averseness of the newsboy can be con-sidered by choosing u(ㆍ) to be increasing and concave. Then the resulting problem can be formulated as a max-imization problem where the objective function represents the expected-utility of the newsboy:

 For any function q(α), which is second order differ-entiable in α if we let ,  respectively, then the first order condition for (4) is

 where α^* is the optimal order quantity which maximizes the expected revenue function of the newsboy.

 As discussed in Eeckhoudt et al. [5], the optimal order quantity, α^*, divides the random yield proportion variable into ranges where an increased order provides a cost or benefit. If we assume y₂＜ ＜ y₁ and a strict concave utility function, u(⋅), we have

 and we have

 To see the effect of the increased risk-aversion on the optimal order quantity, we utilize the concave transformation of the utility function. In general, the increment of the risk aversion is equivalent to a concave transformation as explained in Pratt [14]. Let us assume a concave function k(⋅) with k′(⋅) and k″(⋅). Then E[k(u(Z(Y, α)))] is the objective function of a newsboy who exhibits more risk-averse-ness compared to the newsboy with an objective function as shown in equation (4). In addition, with the strict concavity condition of k(⋅), the following inequality also holds

 where u(Z₀) = u[Z(y₀, α^* )], u(Z₁) = u[Z₁(y₁, α^*)], u(Z₂) = u[Z₂(y₂, α^*)], respectively.

 Denote by H₂(α) = E[k(u(Z(Y, α)))], the first de-rivative of H₂(α) at α = α^* can be shown as following :

 Now, using (5) and (8) the following two inequalities can be derived :

 Denote by β₁, β₂ the right-hand-side of (10), (11) re-spectively, β₁ is always positive (β₁ ＞0) and β₂ is always negative (β₁＞0). Please refer the Appendix for the proofs of (10) and (11). From these we have the following condition

 According to the inequality (12), can take values between a negative number and a positive number. This im-plies that the optimal order quantity of more risk-averse newsboy under uncertain supply environment can be either less of more at the same time when compared to the optimal order quantity of the less risk-averse newsboy.

4. Numerical Study

 The purpose of our numerical study is to verify the results from the previous section. For this purpose we assume the information shown in <Table 1>. The uniform distribution assumption of G(y) might not be realistic. But with this as-sumption we still manage to show the validity of our im-portant findings from Section 3.

<Table 1> Parameter Assumptions

 Under assumptions in <Table 1> if we deploy the equation (3) we can easily show that the optimal number of newspaper which maximizes E(Z(α)] to be α^* = 161.

 Let us now assume that the strict concave utility function u(x) to be －exp(－rx). Futhermore let k(x), the concave transformation function for artificially adding the risk aver-sion, to be－ .

 <Figure 1> shows the optimal order quantities which max-imizes E[u(Z(Y, α)], at various levels of r, the risk aver-sion parameter.

<Figure 1> Optimal Order Quantities for E[u(Z(Y, α)]

 <Table 2> summarizes how the optimal order quantity var-ies as the risk-aversion parameter r increases. In here the optimal order quantity maximizes the expected utility func-tion, E[u(Z(Y, α)]. In contrast to the result of Eeckhoudt et al. [5] the optimal quantities have been increased when the degree of risk-aversion increased.

<Table 2> Optimal Order Quantities for E[u(Z(Y, α)]

 When the level of risk-aversion parameter is small enough(r ≤ 0.0001) the optimal order size is same with the risk-neu-tral solution which maximizes E[Z(α)]. But as the risk-aver-sion characteristic increased (i.e., r increased) the optimal order size rapidly increased to achieve the maximum ex-pected utility. But when the risk-aversion parameter is at its highest level (i.e., r=0.01), it seems that ordering beyond the computed optimal order size could rapidly ruin the news-boy’s expected utility values.

 <Figure 3> illustrates the optimal order quantities max-imizing E[k(u(Z(Y, a)))] where a different kind of utility function, k(u), is introduced. The resulting utility function, k(u(x)), intrinsically exhibit additional risk-aversion level than the previous utility function u(x) = －exp(－rx) via the concave transformation effects.

<Figure 2> Optimal Order Quantities for E[k(u(Z(Y,a)))].

 Surprisingly enough the optimal order size from added risk-aversion, via the concave transformation, is smaller when compared at the same level of risk-aversion parameter values. When r = 0.00001, the optimal order size of our new pro-blem starts at 161 and this is identical with previous optimal order size using u(x) = －exp(－0.00001x) regardless of the concave transformation. But as r increases the optimal order size decreases down to 113. Again at the highest level of risk aversion parameter (i.e., r = 0.01) it seems that ordering below the computed optimal order size could ruin the newsboy’s expected utility values. This is the opposite result compared to the previous case when we do not introduce the concave transformation for introducing the additional risk level or the added risk.

 <Table 3> summarizes optimal order quantities from using objective function E[k(u(Z(Y,a)))].

<Table 3> Optimal Order Quantities for E[k(u(Z(Y,a)))].

 Results presented in <Table 3> together with those in <Table 2> imply many things. Firstly, the added risk aver-sion under the uncertain supply condition plays a role to in-crease the size of the optimal order quantity as shown in the second column of <Table 2>. But the added risk aversion sometimes plays a role to decrease the size of the optimal order quantity as in the second column of <Table 3>. In summary when fore-mentioned two kinds of risk-aversion factors are applied together the optimal order size can possi-bly be greater or smaller than that of low risk-aversion news-boy problem.

5. Conclusion and Future Research

 In this paper we have revisited the famous newsboy prob-lem to develop meaningful insights while the newsboy’s daily newspaper supply is not fixed due to the various reasons. These can include that too many newsboys want to sell news-papers (this situation induces competition on daily newspaper supply) or unexpected poor quality newspapers which can’t be sold to customers. In addition to the supply uncertainty framework we add the risk-aversion into this basic newsboy with uncertain supply to analyze the change of optimal order quantity along the degree of newsboy’s risk-aversion. For setting up the different degree of risk-aversion we introduced utility functions over the revenue function as used in Eeckhoudt et al. [5] and adopted concave transformation on the standard revenue function to produce non-zero positive risk-aversion. From this effort we have found that regardless of the degree of risk aversion the corresponding optimal order quantity is not always greater than that of the risk-neutral newsboy or not always smaller. This implies that the degree of risk-aversion alone cannot make the decision-maker decide decisive actions. In other words the risk-averse newsboy should simultaneously consider other factors such as his retail price, whole price, salvage value, opportunity cost to determine the optimal order quantity.

 Dealing with the same topic under relaxed the known fixed demand assumption can be a challenging future research problem. The added demand uncertainty might result in dif-ferent insight from our findings so far.

 It will also be an interesting future research problem to investigate the exact conditions under which the optimal order quantity of a more risk-averse newsboy is less than that of a less risk-averse one or vice versa.