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I am working on my thesis on revenue management. I have been over the following problem multiple times now, but I fail to see where my mistake is. This example is based on The Theory and Practice of Revenue Management by Talluri and Ryzin, page 61. (See also errata, there is a small mistake in the equation). The recursion is as follows:

$V_t(x)=V_{t+1}(x) + \sum\limits_{j=1}^n \lambda_j(t)(p_j-\Delta V_{t+1}(x))^+$

Where $V_t(x)$ is the value function of capacity $x=0,1,..,C$ and time $t=1,..,T$, the $\lambda_j(t)$ the probability of an arrival of product $j$ at time $t$ and $\Delta V_{t+1}(x)=V_{t+1}(x)-V_{t+1}(x-1)$, the opportunity cost of having one unit of capacity.

The boundary conditions are $V_{T+1}(x)$=0 for all $x$ and $V_{t}(x)=0$ for $t=1,..,T$.

This is a backwards reduction problem, so we start at $t=T$. Since $V_{T+1}(x)=0$ for all $x$, we have $V_T(x)=0 + \sum\limits_{j=1}^n \lambda_j(t) (p_j-(0-0))= \sum\limits_{j=1}^n \lambda_j(t)p_j$ for all $x>0$.

Working back, at $T-1$:

$V_{T-1}(x)=V_T(x) + \sum\limits_{j=1}^n \lambda_j(t-1) (p_j-(V_T(x)-V_T(x-1)))$

Now, since $V_T(x)=\sum\limits_{j=1}^n \lambda_j(t)p_j$ for all $x$, doesn't this reduce to:

$V_{T-1}(x)=V_T(x) + \sum\limits_{j=1}^n \lambda_j(t-1) (p_j-(0))=V_T(x)+\sum\limits_{j=1}^n \lambda_j(t-1)p_j$

I am fairly sure shouldn't happen - why else include the term in the recursion? However, I've been looking at this problem for the past two weeks now and I just really don't get where I am going wrong.

Any help would be greatly appreciated.

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