Production, Manufacturing and LogisticsThe distribution-free newsboy problem under the worst-case and best-case scenarios
Introduction
Since the pioneer works of Arrow et al., 1951, Morse and Kimball, 1951, the classic single-period, single-item inventory problem with random demand, commonly referred to as the newsboy or newsvendor problem, has attracted a great deal of attention and played a central role at the conceptual foundations of stochastic inventory theory; Porteus (2002, chap. 1). Typically, it is formulated as follows. Each day the proverbial newsboy has to decide how many newspapers to stock before observing demand. He purchases them from a publisher at a unit cost c, and sells them at a price p to customers whose uncertain demand is described by a random variable X. Any unsold items are recycled with a unit salvage value s; it is assumed that p > c > s. The newsboy problem is to find the order (purchase) quantity that maximizes the expected profit, and this quantity is known to be the smallest q such that , where F is the cumulative distribution function of X. Numerous extensions of the newsboy problem were reviewed in Khouja, 1999, Qin et al., 2011. Note here that we adopt a classic approach by assuming that the decision-maker is risk-neutral. The models in which he/she is risk-averse, risk-seeking, or uses a maximum entropy approach can be found in Wang et al., 2009, Du et al., 2012, Andersson et al., 2013, Wu et al., 2013.
Scarf (1958) was the first who addressed the distribution-free newsboy problem. He assumed that merely the mean μ = E(X) and the variance σ2 = Var(X) are known, and derived a closed form formula for the order quantity that maximizes the minimum expected profit over all demand distributions with given μ and σ2. For this reason, this worst-case order quantity is also referred to as being found under the maximin criterion. Gallego and Moon (1993) disseminated the rather unnoticed result of Scarf, modified it by restricting the demand distribution to non-negative values, provided a simpler proof with economic interpretations, and showed some applications to other stochastic inventory problems. Alfares and Elmorra (2005) extended the modified Scarf formula to the case of shortage penalty.
Gallego, Ryan, and Simchi-Levy (2001) assumed that X is a discrete random variable taking a finite number of given values. When selected moments and percentiles of X are known, they showed that the worst-case order quantity can be found by solving a linear program. They also demonstrated that if the demand distribution is characterized by μ and σ2 then the maximin policy mostly performs quite well, relative to a policy based on the normally distributed demand. Similar results concerning the performance of the modified Scarf formula were earlier reported by Gallego and Moon (1993). Perakis and Roels, 2008, Andersson et al., 2013 observed, however, that in some situations the maximin policy may lead to an expected profit much lower than the optimal one.
The order quantity found under the worst-case demand scenario is pessimistic (conservative). A less pessimistic policy based on minimizing the expected maximum regret was proposed as an alternative; see e.g. Yue et al., 2006, Perakis and Roels, 2008. Yue et al. (2006) found the corresponding closed form formula for the minimax regret order quantity under the Scarf assumption that μ and σ are known without imposing the non-negativity constraint on the demand distribution. Perakis and Roels (2008) extended their result by adding this obvious constraint, and also solved some cases involving the known median and mode, the symmetricity and unimodality of the demand distribution. Another alternative to the maximin policy was proposed in Andersson et al. (2013). Assuming only the knowledge of μ and σ, the authors demonstrated empirically that finding the most likely distribution in the sense of the maximum entropy leads on average to better results.
To the best of our knowledge, the maximax criterion, which maximizes the expected profit under the best-case demand scenario, has been much less examined. Only trivial cases, resulting from Jensen’s inequality, of the best-case order quantities have been mentioned in Gallego and Moon, 1993, Yue et al., 2006. This observation also refers to other distribution-free stochastic inventory problems discussed in literature; see e.g. Godfrey and Powell, 2001, Wu et al., 2002, Lin and Chu, 2006, Ho, 2009, Kwon and Cheong, 2014.
In this paper we present new theoretical foundations for analyzing the distribution-free newsboy problem under the best-case and worst-case scenarios. Firstly, by using a simple expression for the expected profit, we reveal that this problem actually reduces to the standard newsboy problem with demand distributions that bound the allowable distributions in the sense of increasing concave order. Secondly, we provide a theoretical tool for seeking the best-case and worst-case order quantities when merely the support and the first k moments of the demand are known. Using this tool we derive closed form formulas for such quantities in the case of known support [a, b], mean μ, and variance σ2, i.e. k = 2. Consequently, we generalize all results presented so far in literature for the worst-case and best-case scenarios, and present some new ones. Theoretical tools are also provided for examining the problem when the median, or the mode of the unimodal distribution are additionally available.
The paper is organized as follows. In Section 2 we formulate the problem under study and show its main theoretical result. Section 3 includes a theoretical tool for seeking the sharp lower and upper bounds on the expected met demand needed for deriving the worst-case and the best-case order quantities. Using this tool we find, in particular, closed form formulas for such bounds in the case of known support, mean, and variance. The resulting worst-case and best-case order quantities are presented in Section 4. Section 5 illustrates our findings by the use of a numerical example taken from literature. In Section 6 we present some extensions of our results, while final remarks are made in Section 7.
Section snippets
Problem formulation
In the classic newsboy problem, no cost is assumed if the order quantity does not meet the demand. Although this cost might be difficult to define in practice, we adopt a more general model considered by Alfares and Elmorra, 2005, Perakis and Roels, 2008, in which a known unit lost sales (shortage) cost of ℓ is assumed. Therefore, if q is an order quantity and X denotes the random demand, min(X, q) represents the demand that is met, X − min(X, q) the demand that is unmet, and q − min(X, q) is the
Sharp bounds on the expected met demand
In the previous section we showed that to find the order quantities under the maximin and maximax criteria it suffices to identify sharp lower and upper bounds, L(q) and U(q), on the expected met demand E[min(X, q)]. When only the support and the first k moments of X are assumed to be known, such bounds can be sought by using very general Theorem 2.1 in the monograph of Karlin and Studden (1966, chap. XII). To avoid the use of quite advanced mathematics, below we present a restricted to the
Order quantities
The distributions F and presented in Corollary 1, Corollary 2, Corollary 3 allow us to determine closed form formulas for the corresponding worst-case and best-case order quantities q∗ and . Clearly, using Theorem 1, q∗ () can be defined as the smallest q such that F(q) ⩾ r , where . The corresponding sharp bounds on the maximum expected profits are π(q∗) = (p + ℓ − s)L(q∗) − (c − s)q∗ − ℓμ and .
The order quantity presented below is trivial.
Numerical example
We reconsider the example from Silver and Peterson, 1985, Gallego and Moon, 1993, Alfares and Elmorra, 2005 by letting μ = 900, σ = 122, p = $50.30, c = $35.10, s = $25.00, and ℓ = $14.00. This yields . To specify the support [a, b] of X, which maintains the feasibility of μ = 900 and σ = 122 and allows the comparison of q∗ and with optimal order quantities q∗, we consider some simple cases.
- Case 1.
Let the distribution F of X be uniform on [a, b] with mean μ = 900 and standard deviation σ = 122. Then μ = (
Extensions
Theorem 2 presented in Section 3 can be regarded as a theoretical tool for analyzing the problem when the distribution of demand X is characterized by its support and the first k moments. One can extend this tool by assuming that some additional information about this distribution is also available. As in the case of Theorem 2, the next two results can be deduced from Theorem 2.1 in Karlin and Studden (1966, chap. XII); their proofs are omitted. Theorem 3 Let X have a distribution on [a, b] with known
Final remarks
We presented new theoretical foundations for analyzing the newsboy problem under incomplete information about the probability distribution of random demand. Our main theoretical result stated in Section 2 shows the reduction of the distribution-free newsboy problem under the worst-case and best-case demand scenarios to the standard newsboy problem. We revealed this reduction by finding the relationship between the distribution-free newsboy problem and the notion of the increasing concave order.
Acknowledgements
The author is grateful to two anonymous referees for their critical comments and constructive suggestions that significantly improved the paper.
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