Inventory optimization is one key task in supply chain management. When determining inventory levels, it is necessary to account for the uncertainty of future demand due to forecasting errors and transportation delays. The latter, in particular, is a significant factor in inventory management. Weather and temperature conditions, changes in the natural environment, and political and economic events all contribute to the unpredictability of transportation time. Therefore, it is crucial to consider inventory levels that are resilient against such uncertainties carefully.
When planning inventory for the future, the goal is to cover the demand for some time interval. Due to the possibility of delay, the length of this interval is also uncertain. As a result, future demand to cover follows a mixed distribution, which can be multi-modal; for instance, if the time granularity is weekly, it could be one week’s demand with some probability and, two weeks’ demand with another probability, and so on.
Conventionally, inventory determination sets a deterministic level that makes the out-of-stock probability lower than a predetermined level. However, other methods can result in a lower expectation of inventory level while keeping the expectation of out-of-stock probability below the predetermined threshold. One such method is the randomized inventory strategy. For example, by randomly choosing between two inventory levels, it is possible to achieve the same expected stockout ratio as the deterministic strategy but with a lower expected inventory level. This is not difficult to understand: in randomized strategy, we find the optimal values of the two inventory levels and the optimal assignment probability among them, such that the expectation of inventory is minimized while the expected out-of-stock probability is lower than a predetermined level. The deterministic strategy is a special case of this problem, where these two inventory levels are the same. Since the space solution of the deterministic strategy is a subset of that of the randomized strategy, it is natural that the latter performs not worse than the former.
The key questions, however, are:
How much better is the randomized strategy compared to the deterministic one? How does it depend on the parameters of the problem?
These questions have important implications. In practice, decision-makers might have difficulties understanding the randomized strategy and may be averse to the associated risk. A comprehensive strategy evaluation and a full understanding of its advantages over the deterministic strategy under various scenarios are needed in such situations.
To answer these questions, I first consider a straightforward situation, assuming the delay either does not happen or happens with a fixed length. I compute the expected inventory under various combinations of parameters and compare the randomized and deterministic strategies. In a future post, I will extend the problem set by assuming that the delay follows some more realistic distribution, where the interval of the period can take on multiple values.
A Simple Numerical Example
In this section, we compare deterministic and randomized strategies for setting safety stock, assuming the demand distribution follows a mixture distribution. To simplify, we consider cases where the demand follows this mixture of two normal distributions:
In this setup, the demand Y follows N(μ_h,σ²_h) with probability π ( delay) and N(μ_l,σ²_l) with probability 1−π (no delay). When a delay occurs, the demand interval that needs to be covered increases by (θ−1) times that of the no-delay scenario.
The goal is to set an inventory level at the beginning of the interval such that the expected out-of-stock probability is α.
We consider two strategies:
Deterministic Strategy: Set a fixed level of stock sss such that the expected out-of-stock probability is α:
Randomized Strategy: Randomly choose between two stock levels, s_1 and s_2, with probabilities π′ and 1−π′, respectively. The expected inventory level is π′s_1+(1−π′)s_2. The optimization problem is:
The left-hand side of the constraint contains four cases:
- Safety stock is set to s1s_1s1 while the delay happens.
- Safety stock is set to s2s_2s2 while the delay does not happen.
- Safety stock is set to s1s_1s1 while the delay does not happen.
- Safety stock is set to s2s_2s2 while the delay happens.
We compare the deterministic and randomized strategies under various parameters (θ,σ²_l,α,π,θ). We compute π′,s_1, s_2,s, and the following term as the gain from switching to the randomized strategy for each parameter combination.
We analyze how these quantities change with π and how this relationship shifts with α,σ_l,θ.
Influence of delay probability
Figures 1, 2, and 3 show the behavior of the randomized strategy compared to the deterministic strategy as π, the delay probability changes. The μ_l is set to be 500. θ is set as 1.5. α is set as 0.05. σ_l is set as 15.
When π is very small, the randomized strategy favors s_h, which is close to the s, according to the leftmost panel of Figure 1, the inventory level under the deterministic strategy, and almost ignores s_l (this can be read from the leftmost panel of Figure 2), resulting in minimal gain over the deterministic strategy. This can be seen in the leftmost panel of Figure 3.
As π goes past a certain threshold, as it increases, the deterministic strategy gradually increases the s to keep the out-of-stock probability below the threshold, according to Figure 1. The randomized strategy, on the other hand, sets s_2 slightly higher than μ_l and s_1 slightly higher than μ_h, with a significant probability π′ assigned to s_l, according to Figure 2. This ensures the out-of-stock probability remains low while reducing the expected inventory level.
As π continues to increase, the randomized strategy’s π′ increases gradually, according to Figure 1, with s_1 and s_2 remaining stable, according to Figure. This gradually increases the expected safety stock under the randomized strategy, reducing the benefit over the deterministic strategy, according to Figure 3.
Influence of delay measurement θ
Figures 1, 2, and 3 summarize how different θ values affect the strategies. Figure 3 shows that θ is positively correlated with Δ, indicating a higher θ increases the benefit of the randomized strategy.
Figures 1 and 2 show that a higher θ increases s_1 and s, while s_2 and π′ remain unchanged. As a result, Δ rises.
Impact of Targeted Stockout Ratio α
Figures 4, 5, and 6 show how different α values affect the randomized and deterministic strategies. Figure 6 shows that as α increases, Δ curve shifts upwards, indicating a higher benefit of the randomized strategy.
Close examination of Figures 4 and 5 reveals that a larger α allows the randomized strategy to assign more weight to s_l while keeping the s_h and s_l unchanged. As a result, Δ goes up. This is shown by the downward shift of π′ as α increases.
Influence of σ_l
Figures 7, 8, and 9 summarize how different σ_l values affect the randomized and deterministic strategies. Although there is some instability of curves (probability due to a bad choice of initial guess when solving the constrained optimization), Figure 9 shows that as σ_l increases, the gain from the randomized strategy declines.
This is because larger σ_l values increase s_2, s_1, and s while also increasing π′, which can be read from Figures 7 and 8. This is easy to understand: as σ_l becomes very large, the mixture distribution approximates an unimodal distribution, making the randomized and deterministic strategies equivalent. Thus, the benefit of the randomized strategy diminishes as σ_l increases.
The End
In summary, the randomized inventory strategy can offer significant benefits over the deterministic strategy under certain conditions. The extent of these benefits depends on various factors such as the probability of delay (π), the ratio of demand intervals (θ), the targeted stockout ratio (α), and the variance of the demand distribution (σ_l). Although some conclusions are straightforward, they help us understand more complicated situations, e.g., when delay follows a more realistic distribution. This will be the topic of one of my future posts.
