Randomized Inventory Strategy: When is it effective?(Part 2)

I used a simple numerical example to examine the conditions under which a randomized inventory strategy outperforms a deterministic strategy.

In Part 1, the key takeaway was

• assuming delay follows a Bernoulli distribution, when the probability of delay is negligible, both randomized and deterministic strategies yield similar outcomes.
• -However, when the delay probability exceeds a certain threshold, a randomized strategy — by alternating between small and large inventory levels and putting a considerable weight on the smaller one — can achieve a lower expected inventory level than a deterministic strategy, highlighting the benefits of a randomized approach.
• Beyond this threshold, as the probability of delay increases, the emphasis on smaller inventory levels in a randomized strategy gradually decreases, reducing the randomized strategies’ advantage over a deterministic approach.

The scenario in Part 1 was straightforward: delays either do not occur or occur with a fixed duration, requiring coverage for either a short or long period of demand. In reality, delay times may follow a more complex distribution, such as varying lengths of one week, two weeks, etc., resulting in a mixed distribution of demand to cover. Therefore, it is essential to compare the performance of the two strategies under these more realistic delay distributions, which is the main task of this article.

Set-up

Assume the granularity of time is weekly, and let the delay, denoted by
𝑑, follow a Poisson distribution

The reason for using an integer-based distribution of delay is that, in reality, delays are counted in whole units (e.g., one day, one week). Although the Poisson distribution is not typically used for modeling delays, it serves as a good starting point due to its simple parametric form. Additionally, its variance increases with its mean, which aligns with our intuition: when delays are expected to be significant, we assume they could span many values (in weeks), rather than just a few.

Since both the expectation and variance of a Poisson distribution are equal to λ , a larger λ indicates both a higher mean and higher variance of delay. This property makes the Poisson distribution a practical choice for our model.

When no delay occurs, the demand follows a normal distribution with expectation mu and variance , which covers k weeks demand. Using the above poison distribution, due to the possible delay, the demand to cover follows the following mixed distribution:

The deterministic strategy tries to solve s of the following equation.

The randomized strategy tries to solve s_1, s_2, π′ from the following constrained optimization problem.

Subject to:

and

here, s_1 is the larger inventory level, and s_2 is the smaller inventory level. π′ is the weight put on the larger inventory level.

For other details regarding the randomized and determinstic strategy, please refer to part 1

During the simulation, we plot the behavior of two strategies and the gain from the randomized strategy, defined as

With the distribution parameter λ. We compare their behaviors under various targeted out-of-stock ratios α.

In our experiment, σ² takes value of 20, μ takes 500, k takes 2.

Numerical Results

The result shows some interesting behavior:

1. Cyclical Patterns

The following Figures 1, 2, and 3 illustrate a cyclical pattern in the behavior of the strategies.

Figure 1: The Gain from Randomized Strategy with delay parameter λ, different α

Figure 2: The Behavior of Inventory Levels from Randomized Strategy with delay parameter λ, different α

Figure 3: The weight on larger inventory level of randomized strategy with delay parameter λ, different α

• Figure 2: As λ increases, the two inventory levels (s_1​ for the blue line, s_2 for the red line, and sss for the green line) under the randomized strategy exhibit discontinuous dynamics. They jump to a certain level and remain steady before jumping to a higher level and stabilizing again. The inventory level under the deterministic strategy shows a similar pattern: cyclical jumps with a gradual climb between each jump.
• Figure 3: The weight on the larger inventory level (π′) follows a repeating pattern: it first drops to a low level, gradually climbs, then drops again, and climbs once more.
• Figure 1: Consequently, the gain from the randomized strategy also exhibits a cyclical behavior. It first jumps to a high level, gradually decreases, then jumps to a high level again, and decreases once more.

These dynamics mirror those discussed in Part 1. The reason for this cyclical behavior lies in the nature of the Poisson distribution. The following graph shows the change in probability mass for integers 1–6 as λ increases:

Figure 4: Change of Probability Mass with Lambda for Poisson Distribution

For instance, as λincreases from 0 to 0.74, the weight shifts gradually from 0 to 1 and 2. As λ increases from 0.74 to 1.07, the weight shifts from 0 and 1 to 2 and 3. As λ increases from 1.07 to 1.39, the weight moves from (0, 1, 2) to (3, 4). Although not perfectly precise, this illustrates how the weight consistently transfers from a ‘small side’ to a ‘large side,’ with these sides changing as λ increases. This weight switch is analogous to how we manipulate π, the probability of delay, as discussed in Part 1.

2. Decreasing Gains with Increased Cycles

The gains from the randomized strategy tend to decrease with the number of cycles. This occurs because, as λ increases, the probability mass spreads across many integers. This is evident from the Poisson graph: as x becomes large, the probability mass-λ curve becomes flatter, climbing and decaying more slowly. Consequently, as λ increases, it becomes more challenging to observe a distinct transfer of mass from a ‘small’ side to a ‘large’ side. As a result, the dynamics discussed in Part 1 gradually diminish.

This observation implies the conditions under which a randomized strategy is effective. The distribution should be discrete, with the probability mass concentrated on distinct ‘small’ and ‘large’ sides. This allows the randomized strategy to leverage randomization to outperform the deterministic approach, as explained in Part 1 and this article. Conversely, the randomized strategy may not provide an advantage under a continuous distribution. To verify this, I compared the two strategies, assuming that delay follows a Gamma distribution. The results show that, regardless of the Gamma distribution parameters, the inventory levels are identical for both deterministic and randomized strategies.

I also conducted a simulation to verify the robustness of the observed two patterns, assuming the delay follows a negative binomial distribution. The results of Figure 5–7confirm the robustness of the cyclical behavior under this alternative distribution. (Here, we assume that d follows a the negative binomial distribution with parameter (N,p). We fix the value of p, while changing the value of N to study the behavior of two strategies. The variance and mean of the delay increase with N, and the variance is always 1/p times the mean.)

Figure 5: The Gain from Randomized Strategy with delay parameter N, different α

Figure 6: The Behavior of Inventory Levels from Randomized Strategy with delay parameter N, different α

Figure 7: The weight on larger inventory level of randomized strategy with delay parameter N, different α

Future Directions for Research

There are several possible directions for further investigation:

1. Exploring More Complex Randomized Strategies: To date, we have focused on the simplicity of choosing randomly between two inventory levels. However, future research could explore the effectiveness of selecting from three, four, or even more inventory levels. Although unverified, it is hypothesized that under discrete delay distributions, a strategy involving multiple inventory levels may outperform a two-level strategy, especially when the variance of delay is large. With a high variance, the probability mass is spread across many integers, making it challenging for a two-level strategy to identify distinct ‘large’ and ‘small’ poles. In contrast, strategies with more than two levels could more easily find these distinct poles, potentially offering a significant advantage over deterministic strategies. Future research should investigate the performance of these more complex randomized strategies.
2. Comparing Strategies in Dynamic Settings: Thus far, our analysis has been confined to static settings, focusing on a single period. Since inventory decisions are inherently dynamic, it is crucial to study both randomized and deterministic strategies within a dynamic optimization framework. Examining these strategies in a dynamic context can provide richer insights into how randomization can help reduce inventory levels in more realistic settings. This approach could reveal additional benefits of randomized strategies that are not apparent in static models.

By pursuing these directions, future research can further enhance our understanding of inventory management strategies and their practical applications in various supply chain scenarios.