On the average performance of the adjustable RO and its use as an offline tool for multi-period production planning under uncertainty

Abstract

Robust optimization (RO) is a distribution-free worst-case solution methodology designed for uncertain maximization problems via a max-min approach considering a bounded uncertainty set. It yields a feasible solution over this set with a guaranteed worst-case value. As opposed to a previous conception that RO is conservative based on optimal value analysis, we argue that in practice the uncertain parameters rarely take simultaneously the values of the worst-case scenario, and thus introduce a new performance measure based on simulated average values. To this end, we apply the adjustable RO (AARC) to a single new product multi-period production planning problem under an uncertain and bounded demand so as to maximize the total profit. The demand for the product is assumed to follow a typical life-cycle pattern, whose length is typically hard to anticipate. We suggest a novel approach to predict the production plan’s profitable cycle length, already at the outset of the planning horizon. The AARC is an offline method that is employed online and adjusted to past realizations of the demand by a linear decision rule (LDR). We compare it to an alternative offline method, aiming at maximum expected profit, applying the same LDR. Although the AARC maximizes the profit against a worst-case demand scenario, our empirical results show that the average performance of both methods is very similar. Further, AARC consistently guarantees a worst profit over the entire uncertainty set, and its model’s size is considerably smaller and thus exhibit superior performance.

Appendix

1.1 Demand simulations

In order to examine the performance of the different methods under demand uncertainty we used demand simulations from the uncertainty set. For several uncertainty levels $\rho$, $l=3$ sets of k=100 demand vectors were generated, each vector consisting of T=12 entries to represent a twelve month planning horizon. Each set corresponds to a different distribution, with parameters, that were chosen to explore the effect of symmetric as well as non-symmetric demand distributions.

1.1.1 $U_{box}$

The entries were generated from a linearly transformed Beta distribution with specific shape parameters supported by the uncertainty set for the demand as shown in Table 4. The distributions’ parameters were chosen to explore the effect of symmetric as well as non-symmetric demand distributions.

Table 4 Demand simulations Beta distribution shape parameters

1.1.2 $U_{ellip}$

The demand vectors were generated according to the Generalized Dirichlet distributions (Barthe et al. 2010) on the ball. It is a generalization of the beta distribution to a multivariate distribution. We used the same shape parameter as described in Table 4.

1.2 Cost parameters

The cost parameters are all set relatively to the production cost $c_t$, which was set to 1 for all t. The life-cycle pattern of the demand affects the structure of the selling price vector m. At first, the price is high and it is decreasing according to the life-cycle pattern’s sequential stages. In addition, we assumed that the penalty on shortage p satisfies $p_t=0.5m_t$ for all t. Accordingly, the salvage value was set to $s=m_T=0.8$. These aforementioned cost parameters are identical for all the different sets: C1, C2, and C3. These sets differ in the holding cost vector h. The set C2 has a constant h, which is again set relatively to the production cost vector c. Whereas in sets C1 and C3 we examine the trade-off between the holding and penalty costs by changing the holding cost vector h according to a specific ratio $\frac{h_t}{p_t}$,which was set to 0.1 and 1.5, respectively. Table 5 shows the cost parameters in use. Note, that the initial inventory level $I_0$ was set to zero.

Table 5 Cost parameters