## 3 Vehicle Replacement Optimization Model
### Model Structure  
In the optimization model, car replacement decisions are made based on the cumulated life cycle impact. This approach, for example, allows the assessment of the environmental benefit of vehicle technological improvements in a given decision-making horizon. This concept is illustrated in the schematic example shown in Figure 1.  
The y-axis presents the cumulated environmental impact, e.g., a total ecoinductor score (EI-Pt). The x-axis represents service lifetime or the vehicle kilometers traveled (vkm). For example, the better fuel efficiency of a new car gives tangible environmental benefits compared to the base car (dotted line). Benefits for the environment or carbon credits (hatched area) will accumulate over time. This means that the time a vehicle remains in use should be taken into account in the assessment of environmental impact of vehicle technological improvements. Figure 2 shows a schematic example of the model applied to assessing the benefits of replacing cars. It is assumed that a baseline vehicle is produced and introduced at time t\({}_{0}\). The environmental burdens associated with the production of the baseline/old vehicle are represented by the vertical segment of the blue line. The introduction and operation of the vehicle will accumulate environmental impacts (blue line) that will be minimized throughout the time horizon t\({}_{N}\). The slope of the line of accumulated burden tends to increase with time as a result of the vehicle deterioration expected for older cars. Based on the older cars' deterioration data, at the time t\({}_{A}\) a decision is made to either keep the old car or buy a new one. The vertical segment of the red line at t\({}_{A}\) represents the environmental burdens associated with the production of the new vehicle. Better performance of a new car (red line) will accumulate environmental benefits or emission credits over time. However, making a decision about replacing it requires determining when to retire the old vehicle and how long a new car should be kept to minimize the environmental impact. At t\({}_{A}\), the smaller impact of producing a new car introduces environmental credit (blue arrow with dash line), and the difference in the slope of the cumulative impact of the old and new car accumulates the benefits of using a new car, as explained in Figure 1.  
Figure 1: A schematic example of how environmental impact changes when vehicle fuel efficiency increases. Figure 2: Schematic example of the model applied to assessing the benefits of replacing cars. Optimization of car service lifetime which minimizes the environmental impact needs a balance between the benefits of introducing a new car and its production environmental costs. At time \(t_{B}\), the cumulative benefits from using a new car plus the environmental credit at \(t_{A}\) balance the burden of the manufacturing of a new car. The optimization to minimize the cumulative environmental impact of both cars requires a longer payback period for the new car. With this hypothetical example shown in Figure 2, the minimum lifetime of a new car is \(t_{N}\)-\(t_{A}\). Figure 3 is a schematic example of the optimization model applied to three exchange policies over the given time horizon \(t_{N}\). At time \(t_{A}\), the environmental credit from the replacement of the vehicle (point A) was marked, which, according to Figure 2, is equal to the difference between the cumulative emissions from using an old car and producing the new car. The higher efficiency of the new car will accumulate environmental benefits over time. This is represented by the blue lines coming out of point A. In the first scenario (constant performance decrease rate over time for a new and an old car), cumulated benefits plus the credit in \(t_{A}\) counterbalance production costs (red solid line) over time \(t_{B}\) (point B). But in the longer term \(t_{N}\) (point N), the proceeds from greater efficiency also pay off the credit (dashed red line) from the introduction of a new car in \(t_{A}\).