Management decision-making in the single period optimum pricing problem

Abstract

Management of the firm implementing revenue management has many decisions to make regarding the details of the implementation. In this article we examine seven different decisions surrounding the implementation of aggressive ‘revenue management’ pricing in the context of a firm facing a single period stochastic pricing and stocking problem. We demonstrate through use of an example, that some of the decisions have a large financial impact, while other options that require considerable computational work to implement may provide little financial impact.

Appendix A

The optimum prices for the N-price problem include the deterministic optimum prices for the first N−1 intervals

Consider the case of potentially N different prices during the sales period under additive forecast errors. The firm initially sets a price p ₁ and at the end of the first review period sets a price p ₂, which may be the same as p ₁, (but does not reorder product) and at the second price review point sets a price p ₃ and so on for N potentially different prices during the sales period.

We assume:

• No reordering of product and no holding cost during the period.

• Demand for review period (i) when price p _i is set is independent with pdf f _i (q _i ∣p _i ).

• The probability of the firm stocking out before the last price review point is essentially zero.

For notational convenience, we define the random variable Q _k =∑ _k q _i =∑ _k E(q _i )+∑ _k ξ _i and I _k as the inventory starting the kth interval where I _k =I−Q _k−1>0. We denote the sum of the first k error terms (∑ _k ξ _i ) as X _k and the pdf of Q _k as (.).

Under these conditions, the expected contribution is

p _N is determined at the final price setting point when inventory (I _N ) is known. The necessary condition for a maximum of (A1) w.r.t. p _N is given by (2A) with I=I _N =I−Q _N−1:

We note from (A2) that the value of p _N (=p ^* _N ) that maximizes (A2) is a function of the optimum starting inventory (I ^*) and actual demand for periods 1, 2, …, N−1.

The firm chooses I and p _i , (i=1, 2, …, N−1) to maximize (A1). The necessary condition for a maximum w.r.t. I is that

Noting Q _N−1=E(Q _N−1)+X _N−1, we write

as G[X _N−1, I] and have

since

Now

But ∂p _N /∂I is independent of ξ _N , hence

However, from (A2) the necessary condition for the maximum w.r.t. p _N this becomes

And the necessary condition (A3) becomes

Rearranging

The l.h.s of (A5) is the conditional probability of no stockout given Q _N−1 {or } multiplied by the probability of Q _N−1 and integrated over all values of Q _N−1. It is, therefore, the expected probability of no stockout. We denote this term by F _N (I _N ) where F _N (·) is the cdf of demand in period N.

Necessary condition (A5) then becomes

The necessary condition for a maximum of (A1) w.r.t. p _i i=1, 2, …(N−1) is that

Leading to the necessary condition

Noting ∂p _N /∂ _i does not depend on the error terms ξ _N and X _N−1 (A7) becomes

Making use of condition (A2), this becomes

Using condition (A2) gives

Note that and so (A8) is identical to the necessary condition for deterministic contribution maximization