*2.2. Sales Maximization*

In this section, the optimal number of unit sales for each vehicle model for automobile manufacturers to maximize sales while satisfying the CAFE standards given by Equation (2) is estimated. This is done so by solving the linear programming problem illustrated in Equations (3) through (6) below.

$$\text{Max.} \sum\_{i=1}^{N} p\_i x\_i \tag{3}$$

such that

$$\frac{\sum\_{i=1}^{N} x\_i}{\sum\_{i=1}^{N} \frac{x\_i}{(1+\epsilon)z\_i}} \ge \frac{\sum\_{i=1}^{N} x\_i}{\sum\_{k=1}^{M} \sum\_{i\_k=1}^{N\_j} \frac{x\_i}{z\_k}}\tag{4}$$

$$
\alpha\_i \le \alpha x\_i^\* \tag{5}
$$

$$\sum\_{i=1}^{N} \mathbf{x}\_{i} \le \beta \sum\_{i=1}^{N} \mathbf{x}\_{i}^{\*} \tag{6}$$

where *pi* is the price for each vehicle model, *xi \** is the actual number of units sold, *α* is a parameter for determining the upper limit of vehicle models *i*, *β* is a parameter determining the upper limit of total units sold, and *ε* represents the rate of fuel economy improvement. Equation (4) is a constraint for the linear programming problems in which the relevant company must meet the CAFE standards. In this study, four scenarios are considered: Scenario I, fuel economy for the vehicle models is the baseline value (ε = 1.0); and Scenarios II, III, and IV, in which fuel economy for the vehicle models is uniformly improved from the baseline fuel economy by 10%, 15%, and 20%, respectively (*ε* = 1.1, *ε* = 1.15, and *ε* = 1.2). Next, Equation (5) is the constraint for sales patterns in which the relevant company's current number of units sold for vehicle model *i* grows by a factor *α*, which is set as *α* = 2 for this study. Finally, Equation (6) is the constraint for the total number of units sold, which is set as *β* = 1 for this study. This study solves the sales maximization problem within the four fuel economy improvement scenarios given above (Scenarios I–IV) to estimate the optimal sales pattern for the vehicle models of the relevant automobile manufacturers.

### *2.3. Lifecycle CO2 Emissions of the Automobile Manufacturers*

For gasoline-engine and hybrid vehicle models *i*, the average lifecycle emission intensity per vehicle is found as *fm* by taking the weighted average by number of units sold for the lifecycle emission intensity ( *f g <sup>m</sup>*,*i* and *f h <sup>m</sup>*,*<sup>i</sup>*) derived from the manufacturing, transportation, and sales origin for a single vehicle. Here, one can estimate the lifecycle CO2 emissions (t-CO2) derived from the automobiles as sold by the relevant companies in Japan for 2015 as follows:

$$Q = \sum\_{i \in N\_{\mathcal{S}}} x\_i f\_{m,i}^{\mathcal{S}} + \sum\_{i \in N\_h} x\_i f\_{m,i}^h + \sum\_{i=1}^N x\_i f\_{\mathcal{S},i} + \sum\_{i=1}^N f\_{h,i} \tag{7}$$

where *Ng* is the set of gasoline-engine vehicles models, *Nh* is the set of hybrid vehicle models, *fg,i* is the CO2 emission intensity during travel for vehicle model *i* and *fh,i* is the CO2 emission intensity during disposal of vehicle model *i*.

For a relevant automobile manufacturer, the weighted average fuel economy for a passenger vehicle *i* is defined as as *ei* (km/L) and the lifetime travel distance of passenger vehicles as *d* (km). Thus, *gi* (L), the lifetime gasoline consumption of a passenger vehicle *i*, is obtained as follows:

$$g\_i = \frac{d}{c\_i} \tag{8}$$

The CO2 emissions due to gasoline consumption during travel per vehicle can then be estimated by multiplying the CO2 emission intensity generated per liter of gasoline burned *rg* by the quantity of gasoline consumed *gi* from Equation (6):

$$f\_{\mathcal{S}}^{\text{direct}} = \mathcal{g}\_{i} r\_{\mathcal{S}} = \frac{dr\_{\mathcal{S}}}{\mathcal{e}\_{i}}\tag{9}$$

In addition, the CO2 emissions associated with refining the gasoline necessary for travel per vehicle can be estimated by multiplying the CO2 emission intensity generated per liter of gasoline refined *rc* by the quantity of gasoline consumed *gi* from Equation (8):

$$f\_{\mathcal{J}\_{\mathcal{E}}^{\text{indirect}}}^{\text{indirect}} = g\_i r\_c = \frac{dr\_c}{\mathcal{e}\_i} \tag{10}$$

Thus, the embodied CO2 emission intensity during travel per vehicle *fg,i* in Equation (7) is the sum of *fg,idirect*, the direct emissions generated by gasoline consumption during travel, and *fg,iindirect*, the indirect emissions generated in refining the gasoline:

$$f\_{\mathcal{S},i} = f\_{\mathcal{S},i}^{direct} + f\_{\mathcal{S},i}^{indrect} \tag{11}$$
