Dual-mode inventory management under a chance credit constraint

Abstract

We study a dual-mode inventory management problem of a high-value component where the customer demand and the regular transportation lead time are stochastic, and the review periods of the two modes are different. The manufacturer is subject to a chance credit constraint that bounds the working capital. To solve the resulting chance-constrained stochastic optimization problem, we develop a hybrid simulation optimization algorithm that combines the modified nested partitions method as the global search framework, a feasibility detection procedure for chance constraint verification, and a $\hbox {KN}{++}$ procedure as the final “cleanup” procedure to ensure solution quality. We are then able to analyze the impact of the chance credit constraint on the inventory policies and operational cost. Our numerical study shows that the effects of the reduction in mean or variance of the regular transportation lead time depend on whether the chance credit constraint is loose or tight. We show in this way that this tightness may lead to different mechanisms dominating the observed behavior. Further, we show that substantially extending the deterministic credit limit is less effective than having a slight increase in the probability parameter of the chance credit constraint.

Appendices

Nested partitions method

The nested partitions method is described in Algorithm 4.

Solutions in contour plots

In the following, we show the solutions of the hybrid algorithm in contour plots corresponding to $\alpha = 2\%$ and $\alpha = 5\%$ in Figs. 10, 11, which are similar to Fig. 2 except different locations of the solution points. In addition, we provide the summary of results of the solutions and the corresponding objective function values of the hybrid algorithm in Table 7.

Table 7 Solutions of the hybrid algorithm

Fig. 10

Solutions in contour plots: $\alpha = 2\%$

Fig. 11

Solutions in contour plots: $\alpha = 5\%$

Empirical distribution of the over-limit credit

We further examine the empirical distribution of the “realized credits” in the numerical examples. In particular, we evaluate one optimal solution of our base model (without a credit cap) for $\alpha = 0.01, 0.02, 0.05, 0.10$, respectively, and present the histograms of the over-limit credit (i.e., $\max (-\mathcal {P}(t) - \tau , 0)$) observed in 1000 simulation replications in Fig. 12. It shows that, although our base model does not explicitly enforce a cap on the over-limit credit, the over-limit credit is still bounded to some extent. We further observe that a tighter chance credit constraint results in lower credit line.

Fig. 12

Histograms of the over-limit credit with different $\alpha$ values

The negative cash position is computed according to Eq. (13). The pipeline inventory and on-hand inventory in consecutive periods are correlated. Therefore, the negative cash positions in consecutive periods are also correlated. If the negative cash position is extremely high in one period, it is also likely to be high in the surrounding periods, which altogether increase the overall chance to violate the credit limit $\tau$.