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Abstract
In this paper, we follow a mean-variance (MV) approach to the newsvendor model. Unlike the risk-neutral newsvendor that is mostly adopted in the literature, the MV newsvendor considers the risks in demand as well as supply. We further consider the case where the randomness in demand and supply is correlated with the financial markets. The MV newsvendor hedges demand and supply risks by investing in a portfolio composed of various financial instruments. The problem therefore includes both the determination of the optimal ordering policy and the selection of the optimal portfolio. Our aim is to maximize the hedged MV objective function. We provide explicit characterizations on the structure of the optimal policy. We also present numerical examples to illustrate the effects of risk-aversion on the optimal order quantity and the effects of financial hedging on risk reduction.
Acknowledgements
This research is supported by the Turkish Scientific and Technological Research Council through grant 110M620. We also wish to thank the reviewers for helpful comments and suggestions that improved the content and presentation of this paper.
6 Appendix
In this Appendix, we present some of the derivations and proofs without affecting the flow of the presentation in the main manuscript.
6.1 A Remark on Computational Issues
Computation of the mean function (Equation8(8) ) as well as many other quantities encountered in our analysis requires the marginal as well as conditional distributions. For example, we can write
where fU is the probability density function of U and fK|U = u is the conditional probability density function of K given U = u. Leibnitz rule gives
or
The argument for other random quantities involving the random variables D, K, U, and ST is somewhat similar and one can obtain similar expressions for means, variances, covariances and their derivatives using the marginal and conditional distributions.
6.2 Proof of Theorem 5
Proof. Note that the optimal order quantity y(θ) that maximizes the MV objective is obtained from (Equation18(18) ) by solving m′(y(θ)) − θv′(y(θ)) = 0 for any θ ≥ 0. As a matter of fact, we have already established that the boundary solutions are y(0) = y*RN and y( + ∞) = 0 for θ = 0 and +∞ respectively. For any fixed y, let Θ(y) satisfy the optimality condition (Equation18
(18) ) which can be written as m′(y) − Θ(y)v′(y) = 0 or Θ(y) = m′(y)/v′(y). For all y in the non-dominated region [0, y*RN], we can write
by Lemma 3 and Θ(y) is decreasing in y on [0, y*RN]. Moreover, note that Θ(0) ≥ 0 and Θ(y*RN) = 0 since m′(y*RN) = 0. Therefore, Θ(y) decreases from Θ(0) ≥ 0 to 0 as y increases from 0 to y*RN. By showing Θ(y) is a decreasing function of y on [0, y*RN], we have established the existence of an order quantity y(θ) for each risk-aversion level θ that is between 0 ≤ θ ≤ Θ(0). This solution satisfies m′(y(θ)) − θv′(y(θ)) = 0 or, equivalently, y(θ) = Θ−1(θ). Note that Θ(y) ≤ 0 on (y*RN, +∞) and this region is dominated. Additionally, along the non-dominated region [0, y*RN], we have
which implies that the MV objective function is concave on [0, y*RN]. It is clearly decreasing on (y*RN, +∞) because m(y) is decreasing while v(y) is increasing along this region. Therefore, it is quasi-concave and the order quantity satisfying (Equation18
(18) ) is optimal for (Equation5
(5) ). For any 0 ≤ θ < Θ(0), by taking the inverse Θ−1 of Θ(y), we can obtain the optimal order quantity corresponding to that θ value so that y(θ) = Θ−1(θ). According to the newsvendor’s level of risk-aversion given by θ, the optimal order quantity is between 0 and y*RN. Since θ(y) is decreasing in y, the inverse function Θ−1(θ) is also decreasing. Therefore, it follows that as the level of risk-aversion θ increases the optimal order quantity y(θ) decreases.
6.3 Proof of Theorem 8
Proof. Suppose that there exists an optimal order quantity that satisfies y*RN. Then by concavity of the expected value of the cash flow we know that
and since variance of the cash flow is increasing
From the above arguments we can state that
is dominated by y*RN and this is a contradiction. Therefore, for our analysis we only need to consider the order quantities that lie in the region [0, y*RN].
Note that the optimal order quantity that maximizes the MV objective is obtained from (Equation46
(46) ) by solving
for any θ ≥ 0. As a matter of fact, we have already established that the boundary solutions are
and
for θ = 0 and +∞ respectively. For any fixed y, let Θ(y) satisfy the optimality condition (Equation46
(46) ) which can be written as
We already know from our discussion about the behavior of the mean function in Lemma (3) that in the region [0, y*RN], m′(y) ≥ 0 and m′′(y) ≤ 0. Moreover, via Assumption 7
for all y and
on [0, y*RN] so that for all y in the non-dominated region [0, y*RN] one can show that Θ(y) is decreasing in y. Moreover, note that
and
since m′(y*RN) = 0 by quasi-concavity of E[CF(D, U, K, y)]. Therefore, Θ(y) decreases from θ(0) to 0 as y increases from 0 to y*RN. Up to now, by showing Θ(y) is a decreasing function of y, we establish the existence of an order quantity for each risk-aversion level 0 ≤ θ ≤ Θ(0). Note that on (y*RN, +∞), Θ(y) ≤ 0 which ensures that this is the dominated region. Additionally, along the non-dominated region [0, y*RN] the second order condition is obtained as
Since the second order condition is satisfied, the objective function is concave on [0, y*RN]. Moreover, the first derivative of the MV objective function (Equation46
(46) ) evaluated at y = 0 is
and (Equation46
(46) ) is nonpositive on (y*RN, +∞) because m(y) is decreasing while vα(y) is increasing. Therefore, the MV objective function is decreasing on (y*RN, +∞). This implies that the objective function is quasi-concave and the order quantity that is between 0 and y*RN is a maximizer of (Equation31
(31) ). For any 0 ≤ θ ≤ Θ(0), by taking the inverse Θ−1 of θ(y), we can obtain the optimal order quantity corresponding to that θ value so that
According to the newsvendor’s level of risk-aversion given by θ, the optimal order quantity changes between 0 and y*RN. We show that Θ(y) is decreasing in y so that the inverse is also decreasing. Thus, we state that as the level of risk-aversion θ increases the optimal order quantity
decreases.
6.4 Derivation of (Equation48
(48) ) and (Equation49(49) )
Using the exponential distribution of the demand D with rate λ (which is computationally tractable), we can compute the mean of sales as
Similarly, the second moment is
and variance becomes
Therefore, the mean and variance of the cash flow are
and
Additional information
Notes on contributors
M. Tekin
Süleyman Özekici received his B.Sc. degree in Mechanical Engineering from Boğaziçi University, İstanbul, Turkey, in 1975, and M.Sc. and Ph.D. degrees in Industrial Engineering and Management Sciences from Northwestern University, Evanston, Illinois, in 1975 and 1979, respectively. He held several positions in the Department of Industrial Engineering at Boğaziçi University, İstanbul, Turkey, during the period 1979–2002 and joined the Department of Industrial Engineering at Koç University, İstanbul, Turkey, in 2002. He has also held visiting positions at Northwestern University, the National University of Singapore, and the George Washington University. His research interests focus on the applications of stochastic processes in industrial engineering, operations research, management sciences, and financial engineering.
S. Özekici
Müge Tekin received her B.Sc. degree in Industrial Engineering from Bilkent University, Ankara, Turkey, in 2010 and M.Sc. degree in Industrial Engineering from Koç University, İstanbul, Turkey, in 2012. She is currently a researcher and teaching assistant, working toward a Ph.D. degree in Management Department of Universitat Pompeu Fabra, Barcelona, Spain. Her research interests include pricing and revenue management, operations management and applications of stochastic processes.