**3. Model Preliminaries**

To quantify the impact of operational flexibility on excess inventory, we model the evolutionary dynamics of demand forecasts. There are two types of demand models that can be used for such a purpose: (1) the additive demand model and (2) the multiplicative demand model [12,24]. The *difference* between the successive demand forecasts follows a normal distribution in the additive demand model, whereas the *ratio* of the successive demand forecasts follows a normal distribution in the multiplicative demand model. It has been well established in the literature that the additive demand model fits the empirical data well when the forecast horizon is short and the demand uncertainty is low. However, the multiplicative demand model fits the empirical data well when the forecast horizon is long and the demand uncertainty is high [24]. In this paper, we consider the multiplicative demand model because the lead times are expected to be long when companies source from offshore suppliers. Additionally, we focus on products with high demand uncertainty because the magnitude of excess inventory is more pronounced for products with high demand uncertainty than for those with low demand uncertainty.

We use *D<sup>i</sup>* to denote the demand forecast at time *t<sup>i</sup>* , such that *t*<sup>0</sup> ≤ *t<sup>i</sup>* ≤ *tn*. The forecast-updating process starts at time *t*<sup>0</sup> and ends at *tn*. We fix *t<sup>n</sup>* to the time when the actual demand is realized, so the final demand is fully known at *tn*. Therefore, the length of the forecast horizon is *t<sup>n</sup>* − *t*0. The demand forecasts are updated at each time epoch *t<sup>i</sup>* for *i* ∈ {0, 1, · · · , *n*}. According to the multiplicative demand model, the demand forecast *D<sup>i</sup>* is formulated as follows:

$$D\_i \quad = \ D\_0 e^{\left(\nu(t\_i - t\_0) + \varepsilon\_1 + \varepsilon\_2 + \dots + \varepsilon\_i)}.\tag{1}$$

The *ν* term denotes the drift rate, and the *ε* terms are the forecast adjustments that follow a normal distribution:

$$
\varepsilon\_i \sim \mathcal{N}(-\varrho^2/2, \mathfrak{c}), \quad \forall i \in \{1, \ldots, n\}, \tag{2}
$$

where *ς* is the volatility parameter.

The drift rate can take non-zero values depending on the forecast-updating process. Ref. [12] gives an example of a forecast-updating process such that demand planners use only the advance demand information to update the demand forecasts, which is modeled by a multiplicative demand model with a positive drift rate. When the forecasts are updated based on an unbiased judgmental demand process, the drift rate should be set equal to zero [12]. The multiplicative demand model, given by Equation (1), yields a lognormal distribution for the end demand, which is conditional on the demand forecast at *t<sup>i</sup>* :

$$\ln(D\_n)|D\_i \quad \sim \quad \mathcal{N}(\ln(D\_i) + (\nu - \boldsymbol{\xi}^2/2)(t\_n - t\_i), \boldsymbol{\xi}\sqrt{t\_n - t\_i}), \quad \forall i \in \{0, \dots, n - 1\}. \tag{3}$$

The location parameter of the lognormal distribution is ln(*Di*) + (*ν* − *ς* <sup>2</sup>/2)(*t<sup>n</sup>* <sup>−</sup> *<sup>t</sup>i*), and the scale parameter is *ς* √ *t<sup>n</sup>* − *t<sup>i</sup>* .

In Figure 2, we present an example of the multiplicative demand model with a drift rate of zero and a volatility parameter of one. We normalize the initial demand forecast to one and scale the length of the forecast horizon to one. Thus, *D*<sup>0</sup> = 1, *t*<sup>0</sup> = 0, and *t*<sup>1</sup> = 1. We simulate a random path of the evolution of demand forecasts and calculate the 95% confidence interval over the forecast horizon. The black curve represents the demand forecasts, and the pink area shows the 95% confidence interval. For example, the demand forecast at *t* = 0 is equal to one, and the actual demand is expected to be between zero and four at *t* = 0 given by the limits of the pink area. As shown in the figure, the distance between the limits of the confidence interval decreases over time. This observation indicates that the accuracy of the demand forecasts improves over time as the time for the realization of the final demand approaches, which is consistent with practice. Therefore, the multiplicative model is very effective in capturing the dynamics of demand-updating mechanisms in practice [12,24].

**Figure 2.** Evolution of demand forecasts for the multiplicative process with *D*<sup>0</sup> = 1, *ν* = 0, and *ς* = 1.

We now apply the multiplicative demand model given by Equation (3) to develop the expected profit, optimal order quantity, and expected excess inventory derivations. We consider the classical newsvendor model such that a buyer sells the products in a market with uncertain demand. We use *p* to denote the selling price of a product per unit. The buyer incurs a purchasing cost of *c* per unit. Unsold inventory is salvaged at a salvage value of *s* per unit. The salvage value can be negative in some industries where companies pay to throw away the excess inventory. In the pharmaceutical industry, for example, unsold drugs must be destroyed after their shelf life because of strict regulations, making the salvage value negative for pharmaceutical companies. The critical-fractile solution was developed to determine the optimal order quantity in the classic paper of Arrow et al. [25]:

$$
\beta = \frac{p - c}{p - s},
\tag{4}
$$

where *β* is known as the critical fractile or the critical ratio. When the demand follows the lognormal distribution given by Equation (3), the optimal order quantity is found by:

$$\begin{array}{rcl} \mathcal{Q}^\* &=& \mathcal{e}^{\ln(D\_{\bar{i}}) + (\nu - \mathfrak{e}^2/2)(t\_{\mathfrak{n}} - t\_{\bar{i}}) + \Phi^{-1}(\mathfrak{k}) \,\, \mathfrak{g} \, \sqrt{t\_{\mathfrak{n}} - t\_{\bar{i}}}} \end{array} \tag{5}$$

where Φ−<sup>1</sup> (·) is the inverse of the standard normal distribution function Φ(·).

To find the expected profit, we first need to derive the standardized order quantity. When the buyer orders *Q* units, the standardized order quantity becomes:

$$z\_{\mathcal{Q}} = \frac{\ln(\mathcal{Q}/D\_i) - (\upsilon - \mathfrak{g}^2/2)(t\_n - t\_i)}{\mathfrak{g}\sqrt{t\_n - t\_i}}.\tag{6}$$

Then, the expected profit for an order quantity of *Q* units is given by Bicer and Hagspiel [10]:

$$E(\Pi(Q)|D\_i) = (p-c)Q - (p-s)\left[Q\Phi(z\_Q) - D\_i e^{\nu(t\_\pi - t\_i)}\Phi(z\_Q - \zeta\sqrt{t\_\pi - t\_i})\right].\tag{7}$$

The first term on the right-hand side of Equation (7) gives the total profit when all the units ordered are sold in the market at the selling price. However, the demand is uncertain, and it can be less than *Q* units. The second term on the right-hand side of the expression can be considered as the cost of an insurance policy that fully hedges the excess inventory risk. The term in brackets is the expected excess inventory:

$$\mathbb{E}\left(\text{Excess Inventory} \mid \text{Q}\_{i}\text{D}\_{i}\right) \quad = \; Q\Phi(z\_{\text{Q}}) - D\_{i}e^{\nu(t\_{\text{n}}-t\_{i})}\Phi(z\_{\text{Q}} - \zeta\sqrt{t\_{\text{n}}-t\_{i}}).\tag{8}$$

The last expression indicates that the excess inventory (hence the waste) can be reduced using two different approaches. First, postponing the ordering decision leads to a reduction in the time window *t<sup>n</sup>* − *t<sup>i</sup>* , which in turn helps decrease the expected excess inventory. Second, reducing the order quantity results in a decrease in the expected excess inventory.

These results provide useful insights regarding the use of operational flexibility to improve sustainability by reducing waste in the sourcing process. The lead-time reduction and quantity-flexibility practices make it possible for the buyer to postpone their ordering decision. Therefore, these two operational-flexibility strategies help decrease excess inventory. Utilizing multiple sources (one offshore supplier and one domestic supplier), the buyer can reduce the quantity ordered from an offshore supplier. Thus, multiple sourcing also helps reduce the excess inventory.
