# Enabling Horizontal Collaboration in Logistics Through Secure Multi-Party Computation
## 3 Optimization Problem
### On the Potential of Horizontal Collaboration  
When aiming for a privacy-preserving solution enabling competing hauliers to jointly assign their orders to companies, a natural question to ask is whether this actually brings any benefit regarding the optimization goal in Equation (1). The answer to this question can be seen in Figure 1, derived from [3]: for different instance sizes, the left (blue) box marks the objective value if each haulier optimizes locally, while the right (orange, dotted) box marks the objective value if the same orders are processed in a collaborative manner. These results underpin the intuition that a collaborative approach, where empty space on one truck can be used for an order from another haulier, increases the efficiency and optimality of the entire solution. Note that this collaborative heuristic not only optimizes Equation (1) but also significantly contributes to all the dimensions listed in Section 1, underscoring its potential for sustainable logistics solutions.  
An important observation is revealed by a closer look at the figure. Namely, collaboration pays off more for instances with fewer orders, an aspect that is also reflected in the diminishing differences in the collaborative and concurrent objective values as the number of orders increases. This is not surprising, since higher numbers of orders per haulier already allow for reasonably efficient in-house optimization.  
This observation can now be used to further incentivize hauliers to participate in a CTN through the following approach:  
* In a first step, each haulier performs in-house optimizations, maximizing the load factor of their own trucks, which will result in good results for most of the truck drives if they have a sufficiently large number of orders to process.
* Only in a second step, orders that cannot be efficiently processed in-house--e.g., because they are based in outlier destinations, or because they just do not fit on the optimized trucks--are submitted to the collaborative platform, together with available trucks. The benefits of this two-step approach are now twofold:
* From a company point of view, this allows one to hide the actual order situation (being sensitive economic information) as only \"left-over\" orders are provided to the system, and no information about pre-processed orders is revealed.
* From an optimization point of view, this brings the advantage of reduced instance sizes given the NP-hardness of the CFDP and the additional overhead of cryptographically secured implementations.  
In particular, looking at Figure 1, instances with about 500 orders already achieve a very good overall load factor (<12% of empty deck space), so that not too much additional  
Figure 1: Comparison of objective value in collaborative and competitive settings. Displayed figures are obtained across multiple runs and instances. Data derived from [3; 5].  
improvements are possible. Therefore, scalability in terms of the number of orders is only needed for up to several hundred, or at most 1000, orders. In case a higher number of orders is submitted to the platform by different hailiers, these can be processed independently in (randomized) batches of 500 to 1000 orders, without relevant negative impact on the quality of the solution.