*4.3. Multiple Sourcing*

In the lead-time reduction and quantity-flexibility practices, the buyer can source products from only one supplier. In the former case, the buyer can source products from either an offshore supplier or a domestic supplier but not from both at the same time. In the latter case, the buyer can only order from an offshore supplier that provides the flexibility to adjust the initial order quantity in a later time epoch. We now consider an alternative strategy, whereby the buyer can source products from two different suppliers at the same time: One is the offshore supplier and the other the domestic.

The multiple sourcing strategy is very effective in mitigating the risk of supply– demand mismatches [9]. By utilizing an offshore supplier, the buyer benefits from the cost advantages of offshore production. If the buyer orders lower quantities from the offshore supplier, the excess inventory risk can also be minimized. If the demand turns out to be unexpectedly high, the buyer then utilizes the domestic supplier to meet the surplus demand. Therefore, a multiple-sourcing strategy allows the buyer to benefit from the cost advantages of the offshore supplier and the responsiveness of the domestic supplier at the same time. However, one of the implementation challenges of this strategy is that the domestic supplier may not always be utilized at a high level. If the demand turns out to be low, the quantity ordered from the domestic supplier would not be high enough to fully utilize its available capacity. Therefore, domestic suppliers are exposed to the risk of capacity underutilization when a multiple-sourcing strategy is employed. To compensate for this risk, domestic suppliers often charge their buyers a capacity reservation fee.

To capture these dynamics, we consider a multiple-sourcing setting with one buyer, one offshore supplier, and one domestic supplier. The buyer determines the order quantity of *Q<sup>l</sup>* units from the offshore supplier and reserves a capacity of *K* units with the domestic supplier at time *t<sup>l</sup>* . We use *c<sup>l</sup>* and *c<sup>k</sup>* to denote the cost of ordering from the offshore supplier and the capacity reservation cost at the domestic supplier per unit, respectively. At time *t<sup>n</sup>* > *t<sup>l</sup>* , the buyer observes the final demand and determines the final order quantity of *Q<sup>s</sup>*

units from the domestic supplier such that *Q<sup>s</sup>* ≤ *K*. The domestic supplier is additionally paid *c<sup>s</sup>* per each unit ordered. The formulation of *Q<sup>s</sup>* is given by Biçer [9]:

$$Q\_s = -\max(\min(D\_{\text{tr}}Q\_l + \text{K}), Q\_l) - Q\_{l\prime} \tag{13}$$

where *D<sup>n</sup>* is the final demand for the product, which is observed at time *tn*. Following up from Biçer [9], we first define two ratios to derive the optimal values of *Q<sup>l</sup>* and *K*. They are:

$$
\beta\_1 \quad = \quad \frac{c\_s + c\_k - c\_l}{c\_s - s} \, \tag{14}
$$

$$
\beta\_2 \quad = \quad \frac{p - c\_s - c\_k}{p - c\_s}.\tag{15}
$$

Then, the optimal values are given by Biçer [9]:

$$\mathbf{Q}\_{l}^{s} \quad = \quad \mathbf{D}\_{l} \mathbf{e}^{(\nu - \zeta^{2}/2)(t\_{n} - t\_{l}) + \Phi^{-1}(\beta\_{1})\xi\sqrt{t\_{n} - t\_{l}}} \, \tag{16}$$

$$K^\* = -D\_l e^{(\nu - \zeta^2/2)(t\_n - t\_l)} [e^{\Phi^{-1}(\beta\_2)\varsigma\sqrt{t\_n - t\_l}} - e^{\Phi^{-1}(\beta\_1)\varsigma\sqrt{t\_n - t\_l}}].\tag{17}$$

Then, the profit in the multiple-sourcing setting is formulated as follows:

$$\Pi(Q\_l, \mathcal{K}, Q\_s) = p \min(D\_{\mathcal{W}}, Q\_l + \mathcal{K}) + s \max(Q\_l - D\_{\mathcal{W}}, 0) - c\_l Q\_l - c\_k \mathcal{K} - c\_s Q\_s. \tag{18}$$

Using this formula and simulating the demand paths, the expected inventory for the optimal *Q<sup>l</sup>* and *K* levels can be found. The expected excess inventory can also be calculated by plugging *Q<sup>l</sup>* into Equation (8).

We now present an example to demonstrate the impact of multiple sourcing on the buyer's profits and excess inventory. We assume the same demand parameters as the examples given above. The cost of purchasing from the offshore supplier is USD 40 per unit, and the cost of purchasing from the domestic supplier is USD 50 per unit. The domestic supplier also charges USD 15 for each unit of capacity reserved. Unlike the lead-time reduction and quantity-flexibility examples, we cannot vary the key decision parameter that determines the magnitude of operational flexibility in the multiple-sourcing setting. In the multiple-sourcing setting, operational flexibility is directly influenced by the reactive capacity *K*. However, the reactive capacity is not a control variable. It is a decision variable that has to be optimized depending on the cost and demand parameters. For that reason, we vary the selling price between USD 150 and USD 300 in our analysis because Equation (17) indicates that the optimal capacity level increases with the selling price. In other words, we can observe the impact of responsiveness on profits and the waste ratio by varying the selling price because the selling price is directly correlated with the capacity level.

Figure 5a shows the impact of the selling price on the percentage profit increase such that the profit increases with the selling price. Doubling the selling price from USD 150 to USD 300 increases the profit by almost 190%. Figure 5b demonstrates that the waste ratio does not change depending on the selling price. This is because the optimal order quantity from the offshore supplier does not depend on the selling price but only on the cost parameters. For this reason, there is no environmental benefit to improving operational flexibility with the multiple-sourcing strategy. Both the profit increase and waste ratios in Figure 5a,b are calculated with respect to the resulting profit and waste when the selling price is set at USD 150.

We summarize the results of the three operational-flexibility strategies in Table 1 below. For each strategy, the profit increase and waste ratio reported are calculated by setting the parameter of the strategy in the table to the maximum value in comparison to setting the parameter to the minimum value within the given range. The results show that leadtime reduction has the highest potential in reducing waste while improving the profits of companies. In Section 5, we further elaborate on and discuss the results of the three strategies.

**Table 1.** Summary of results of the operational-flexibility strategies.

