One of the most recent extensions of the classical newsvendor problem (NVP) is the formulation of a bicriteria decision problem whereby the newsvendor incorporates into the same objective function the conflicting goals of maximizing the expected profit and the probability of exceeding it, the latter known also as the satisfying objective. Parlar and Weng (2003) review the literature on the use of the two objectives within the NVP framework and present the pioneering work on the subject. Arcelus et al (2012) obtains a closed form solution for a uniformly distributed demand, but with a constant price. This is done by using the existence of a closed form solution for the second criterion to arrive at the solution for the Bicriteria problem. The other promising research area in NVP framework is to model both the price and order quantity in the retailer’s optimization policy and follows the pioneer work of Petruzzi and Dada (1999).
In this paper, we have merged these two extension areas and address the bicriteria decision problem whereby the retailer incorporates in its objective function the conflicting goals of maximizing the expected profit and the probability of exceeding it and determines both the optimal price and the order quantity within a newsvendor framework.
The Bicriteria Model
The model starts by considering a retailer who must set a retail price p, and acquire Q units of a given product, to meet a uniformly distributed demand, x, with finite support [g-u/2,g+u/2], a mean of g(p), a standard deviation of σ, a density function of f(x) and a cumulative distribution function of F(x). The purchasing cost per unit is c. Any excess inventory can be disposed of at a price of v, with Λ(Q) as the expected leftovers. Any unfulfilled demand unit incurs a penalty of s, with Θ(Q) as the expected shortages. Define Π(p,Q) as the retailer’s profit function and as the retailer’s actual profit, for a given pair of values of the order quantity and the realized demand. Also, let E(p,Q) be the expected profit, and H(p,Q), the probability of the profit achieving its expectation. The expressions for these functions are given by:
(1)
To derive the expression for the bicriteria index, Y(p,Q), let E* and H* be, respectively, the optimal value of E and H, both obtained below through the separate maximization of E and H, with respect to p and Q. Further, let be the corresponding optimal retail price, order quantities, expected shortages and expected leftovers for each goal i. Then, the bicriteria problem consists of finding the retail price,
, and the order quantity,
, that maximizes the bicriteria index Y(p,Q), where
(2)
Y(p,Q) consists of the weighted average of the indices representing the extent to which E and H differ from their respective optimum (E* and H*), weighted by their relative importance as measured by w and 1-w. Observe that both E* and H* are constants in the bicriteria function of (2). Their role is to normalize the performance measure for each goal, thereby removing the units of measurement and thus, rendering them comparable.
Given the complexity of the problem, a numerical section highlights the main features of the model. The linear / iso-elastic demand functions and additive / multiplicative error structures are considered while illustrating the managerial insights.
References
- Arcelus, F. J., Kumar, S., and Srinivasan, G., 2012. Bicriteria optimization in the newsvendor problem with uniformly distributed demand. 4OR – A Quarterly Journal of Operations Research, 10(3), 267-286.
- Parlar, M., Weng, Z. K., 2003. Balancing desirable but conflicting objectives in the newsvendor problem. IIE Transactions, 35, 131-142.
- Petruzzi, N. C., Dada, M., 1999. Pricing and the newsboy problem: A review with extensions. Operations Research, 47, 183-194.