*4.2. Quantity-Flexibility Contracts*

When the buyer does not have the possibility of implementing local sourcing, a quantity-flexibility contract can be used to increase profits and reduce excess inventory. Under a quantity-flexibility agreement, the buyer determines the initial order quantity at time *t<sup>l</sup>* and the flexibility percentage. We use *Q<sup>l</sup>* to denote the initial order quantity and following the terminology in [10] we use *α* to denote the flexibility percentage. Then, the buyer determines the final order quantity *Q<sup>f</sup>* at time *t<sup>s</sup>* > *t<sup>l</sup>* within some limits:

$$(1 - \mathfrak{a})Q\_l \le Q\_f \le (1 + \mathfrak{a})Q\_l. \tag{9}$$

If the demand forecasts are updated upward from *t<sup>l</sup>* to *t<sup>s</sup>* , the buyer would increase the order quantity up to (1 + *α*)*Q<sup>l</sup>* units. Otherwise, the buyer would decrease the order quantity down to (1 − *α*)*Q<sup>l</sup>* units. Thus, the final order quantity depends on the demand forecast at time *t<sup>s</sup>* , the initial order quantity, and the flexibility percentage. It is given by [10]:

$$Q\_f = \begin{cases} Q\_l(1-\mathfrak{a}) & \text{if } \quad D\_{\mathfrak{s}} < D\_{\mathfrak{s}1} \\ Q\_f^\* = D\_{\mathfrak{s}} e^{(\nu - \mathfrak{s}^2/2)(t\_\mathfrak{n} - t\_\mathfrak{s}) + \mathfrak{a}^{-1}(\mathfrak{f})\xi\sqrt{t\_\mathfrak{n} - t\_\mathfrak{s}}} & \text{if } \quad D\_{\mathfrak{s}1} \le D\_{\mathfrak{s}} \le D\_{\mathfrak{s}2} \\ Q\_l(1+\mathfrak{a}) & \text{if } \quad D\_{\mathfrak{s}2} < D\_{\mathfrak{s}1} \end{cases} \tag{10}$$

where:

$$D\_{s1} = Q\_l(1 - \alpha)e^{-(\nu - \zeta^2/2)(t\_n - t\_s) - \Phi^{-1}(\beta)\xi\sqrt{t\_n - t\_s}}\tag{11}$$

$$D\_{s2} = Q\_l(1+a)e^{-\left(\nu-\zeta^2/2\right)(t\_n-t\_s)-\Phi^{-1}(\mathcal{G})\zeta\sqrt{t\_n-t\_s}}.\tag{12}$$

The *Ds*<sup>1</sup> and *Ds*<sup>2</sup> terms can be interpreted as the lower and upper critical values for the demand forecast at time *t<sup>s</sup>* . If the demand forecast *D<sup>s</sup>* turns out to be higher than *Ds*2, the buyer should order the maximum allowable quantity based on the quantity-flexibility contract, which is equal to *Q<sup>l</sup>* (1 + *α*) units. If the demand forecast *D<sup>s</sup>* turns out to be lower than *Ds*1, the buyer should reduce the order quantity to the minimum allowable level, which is equal to *Q<sup>l</sup>* (1 − *α*) units. If the demand forecast *D<sup>s</sup>* is between these limits, the buyer should set the order quantity to the profit-maximizing level. Based on the final order quantity, the expected profit and the expected excess inventory can be calculated by Equations (7) and (8), respectively.

In Figure 4, we present an example of a buyer who orders products from an offshore supplier and has the flexibility to update the initial order quantity based on a quantityflexibility contract. The cost parameters are the same as above: The selling price is USD 300 per unit, the cost of purchasing from the offshore supplier is USD 40 per unit, and there is no salvage value for unsold inventory. Likewise, the demand parameters are the same as above. The demand forecast at *t*<sup>0</sup> is normalized to one. The drift rate and the volatility are equal to zero and one, respectively. The initial order quantity is determined at the very beginning such that *t<sup>l</sup>* = 0. The final order quantity is determined at *t<sup>s</sup>* within the quantity-flexibility limits.

Figure 4a shows the impact of flexibility on the percentage profit increase. The x-axis represents the flexibility percentage *α*, and the y-axis represents the percentage increase in profits as a result of the order-adjustment flexibility. To calculate the values of the profit increase, we generate 100,000 random demand paths for each *α* value. We compare the demand realization at *t<sup>s</sup>* along each sample path with *Ds*<sup>1</sup> and *Ds*<sup>2</sup> limits to determine the final order quantity. Then, the expected profit is calculated using Equation (7). Figure 4a demonstrates that the percentage change in profit increases with a decreasing rate as the flexibility increases. When *α* = 0.4, the buyer can achieve around 20% profit increase compared with the no-flexibility case.

**Figure 4.** Analysis of quantity flexibility: (**a**) Impact of the flexibility parameter on profit increase; (**b**) Impact of the flexibility parameter on waste ratio.

Figure 4b depicts the waste ratio as a function of the flexibility parameter. We calculate the waste ratio as the ratio of the expected excess inventory when the buyer has the flexibility to update the initial order quantity to the expected excess inventory when the buyer has no flexibility. Therefore, the waste ratio is close to one when the flexibility percentage is near zero. As the flexibility percentage increases, the waste ratio decreases with a decreasing rate. The curve becomes flatter for high *α* values such that the waste ratio cannot be reduced below 40%. These results indicate that quantity flexibility has positive economic and environmental impacts on the sourcing process of the buyer. However, its environmental impact is more limited than what can be achieved with lead-time reduction.
