Model retraining and information sharing in a supply chain with long-term fluctuating demands

We consider a simple SC consisting of $N$ echelons adopting an order-up-to-point policy Chen2000a ; Lee2000 ; Gilbert2005 (Fig. 1). Each echelon may represent a retailer, wholesaler, or manufacturers of certain consumer goods, for example. When goods are sold at the consumer end of the SC (i.e., retailer), its inventory is reduced, and orders are placed with a wholesaler to replenish the products. The wholesaler places orders with the next wholesaler or manufacture to meet the demand. As the result of these orders, products or materials are sent in the opposite direction.

The target inventory level in the policy is determined by the forecasting model of each echelon, which is described in Sec II.2. Briefly, each echelon first receives an order from a downstream echelon (or the end customer) and performs shipments to it, followed by an immediate replenishment process by sending an order to the next upstream echelon, in a single discrete time step. A single discrete time step represents an interval between successive orders, which may represent a day, week, and so forth, in a real situation. For simplicity, here, we assume that each firm in the SC places an order in a synchronized manner in each time step. Note that we ignore the lead time in this model to exclude its effects on the bullwhip effect in our model.

Technically, we update the system from $t=T$ to $t=T+1$ as follows. First, at the most downstream customer, demand is generated by a Gaussian distribution, $\mathcal{N}(\mu_{0}(T),\sigma_{0}(T))$ (with temporally variable parameters; see Sec. II.3 for details). Note that we have carried out the entire simulations with log-normal distributions as well (Supporting Information) to confirm the conclusions were not specific to the Gaussian distribution. Then, we sequentially update each echelon from the downstream. An update of an echelon is twofold, involving a shipment to the next downstream echelon and subsequent replenishment by placing an order to the next upstream echelon. Echelon $i$ receives an order from the next downstream echelon $i-1$ (or the end customer for $i=1$), which is denoted by $O_{i-1}(T)$. If the ordered amount is smaller than the inventory level of echelon $i$, $I_{i}(t)$ (i.e., $O_{i-1}(T)<I_{i}(T)$), echelon $i$ depletes the ordered amount (i.e., $I_{i}(T+0.5)=I_{i}(T)-O_{i-1}(T)$); otherwise, it depletes the entire current inventory (i.e., $I_{i}(T+0.5)=0$). The downstream echelon $i-1$ receives this shipment from echelon $i$, $S_{i}(T)=\max{\{O_{i-1}(T),I_{i}(T)\}}$, resulting in $I_{i-1}(T+1)=I_{i-1}(T+0.5)+S_{i}(T)$. For $i=N$, the ordered amount is replenished (i.e., $I_{N}(T+1)=I_{N}(T+0.5)+O_{N}(T)$). Note that we do not consider the backlog of orders; thus, the amount of demand that exceeds the inventory level is lost, which is recorded as the lost sales opportunity.

Each echelon $i$ has a forecasting model for demand as a probability distribution, based on which it determines the safety stock and target inventory levels. Here we use the Gaussian distribution, $O_{i-1}(t)\sim\mathcal{N}(\mu_{i}(t),\sigma_{i}(t))$, as the forecasting model. This model is one of the simplest forecasting models and mimics the functions of other types of more complex models that generate a prediction with an estimated error. We use this model to exclude the possibility that a specific temporal structure of a forecasting model, not the retraining of the models on which we focus here, causes the bullwhip effect. Essentially the same models for the demand have been widely used in the literature to theoretically investigate the dynamics of SCs Cachon2000 ; Raju2000 ; Cachon2001 ; Kim2006 .

Using the two parameters in the model (i.e., $\mu_{i}$ and $\sigma_{i}$ denoting the mean and standard deviation, respectively), the target inventory level is set to

where $c$ is the safety factor defining the amount of safety stock (i.e., order-up-to-point policy Chen2000a ; Lee2000 ; Gilbert2005 ). The ordered amount is simply the difference between the target and current (at $t=T+0.5$) inventory levels (i.e., $O_{i}(T)=I_{i,\rm{target}}(T)-I_{i}(T+0.5)$). If the current inventory level is higher than the target inventory level (which may occur when the forecasting model is retrained), we set $O_{i}(T)=0$. Note that this order policy alone does not amplify the demand signal, and thus does not cause the bullwhip effect Ren2017 .

We also fluctuate the demand distribution at the end customer, $\mathcal{N}(\mu_{0},\sigma_{0})$. As an example, we set the following simple stochastic process with moderate variability. We fixed $\sigma_{0}=0.1$. When updated, the mean of the distribution, $\mu_{0}$, was redrawn from a uniform distribution between $1/2$ and $2$, i.e., $\mu_{0,{\rm new}}\sim U(1/2,2)$.

We considered two types for intervals of updating the distribution, $L_{\rm{int}}$. The first is a constant update interval, with which the demand distribution is updated every $L_{\rm{int}}(=Const.)$ time steps. In this study, we examined $L_{int}=50$ and $100$. The second type of interval is a random update interval, which is redrawn from the uniform distribution, $U(L_{\rm{min}},L_{\rm{max}})$, every time the demand distribution is updated. Here, we set $L_{\rm{min}}=50$ and $L_{\rm{max}}=100$. These two update intervals were used to show that our results were not influenced by either discrete nor stochastic properties of the demand.

As the demand pattern varies, the forecasting models of the echelons are also updated (i.e., retrained). Each echelon $i$ refers to the past $L_{\rm{train}}$ orders it received, $\{O_{i-1}(T-1),\ldots,O_{i-1}(T-L_{\rm{train}})\}$. When the model is updated, $\mu_{i}$ and $\sigma_{i}$ were replaced by the mean and corrected standard deviation computed from the sample, respectively. The optimal value of $L_{\rm{train}}$ depends on the environment and is generally unknown.

We consider the following four types of schemes defining when and how to perform retraining at each echelon.

We performed simulations for $N=5$ echelons, varying $L_{\rm train}$. The safety factor was set to $c=1.96$. As initial conditions, we set $\mu_{0}=\mu_{1}=\cdots=\mu_{N}=1.0$, $\sigma_{0}=\sigma_{1}=\cdots=\sigma_{N}=0.1$, and $I_{1}=\cdots=I_{N}=1.196$.

To evaluate the efficacy of the retraining schemes, we prepared a baseline model, which is referred to as the constant policy. In this policy, each echelon’s prediction is based on the Gaussian distribution, $\mathcal{N}(\bar{\mu},\sigma)$, which does not change over time. Here, $\bar{\mu}=1.25$ is a fixed parameter representing the long-term mean of demands ($\mu_{0}\sim U(1/2,2)$), and we examined various values of $\sigma$ ranging from 0 to 1.2 in separate runs. This policy assumes an ideal situation in which each echelon knows the value of $\bar{\mu}$.

First, we present the average inventory level of each echelon in Fig. 1. Upstream echelons had more inventory than downstream echelons except when the shared forecasting model algorithm was applied. A higher inventory level indicates that the echelon estimates larger variability in the demand signals (i.e., larger $\sigma_{i}$), resulting in ordering more to have more stock.

With the independent update algorithm, the inventory level became extremely high in upstream echelons when $L_{\rm train}$ was small (Fig. 1(b)). In this case, unnecessarily frequent retraining perturbed the system too much, and its amplification caused the bullwhip effect.

As expected, the shared forecasting model algorithm had the best performance in this respect. Because the forecasting model was shared to upstream echelons, they had the same estimate of demand variability, resulting in identical inventory levels.

We then examined the level of service achieved. In general, a lower inventory level leads to more stockouts and subsequent loss of sales opportunities. Even if we consider the backlog of orders, stockouts causing additional costs are undesirable Croson2003 ; Disney2007 . We quantified the number of lost sales opportunities of echelon $i$ at $t=T$ by $O_{i-1}(T)-S_{i}(T)(=\max{\{0,O_{i-1}(T)-I_{i}(T)\}})$. This value was divided by 1.25 (i.e., the long-term average of the demands) to compute the percentage of the lost sales opportunities (Fig. 2).

The regular update scheme (Fig. 2(a)) caused a substantial number of sales losses, which reached the maximum around $L_{\rm train}=T_{\rm{int}}/2$ where the inventory level was at its minimum (Fig. 1(a)).

The independent update scheme (Fig. 2(b)) suppressed the sales losses to a reasonable level while keeping the inventory level moderate when $L_{\rm{train}}$ was not very small (Fig. 1(b)).

The shared forecasting model scheme significantly reduced the sales losses (Fig. 2(c)) with a low inventory level (Fig. 1(c)).

In Fig. 2(a) and (b), echelon 2 or 3, not echelon 1, had the most sales losses. This is explained as follows. In these schemes, echelon 1 swiftly adjusted to the change in demand, and the upstream echelons followed after some time. When the demand increased, because echelons 2 and 3 did not receive an order larger than $I_{1}$, which is smaller than the inventory level of the other echelons, the increase in demand became difficult to detect, and adaptation to it was delayed.

To further evaluate the performance of the three schemes (i.e., regular update, independent update, and shared forecasting model schemes), we consider a two-dimensional plane of the total inventory level (i.e., sum of $I_{i},$ $i=1,...,N$) in the system, and the percentage of lost sales opportunities at echelon 1 (Fig. 3). Note that the lost sales at echelon 1 are equal to those of the entire SC. Because these two indices have in a trade-off relationship, we must evaluate both at the same time to measure the performance. The lower left area in Fig. 3 corresponds to a lower inventory level and fewer lost sales opportunities (i.e., a set of better operating points). For various values of $L_{\rm train}$, the results of the three retraining schemes are plotted with the results of the constant policy (red dashed line). Symbols located above this line indicate that the inventory control is inferior to that of the constant policy.

The performance of the regular update scheme was comparable to that of the constant policy. In addition, the independent update and the shared forecasting model schemes provided substantially better operating points than the constant policy. In particular, the shared forecasting model scheme had the best performance. Moreover, the operating points of the shared forecasting model scheme were distributed over a small area, indicating the robustness of the scheme.