# Solving vehicle assignment problems by process-network synthesis to minimize cost and environmental impact of transportation

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## Abstract

A method and software are proposed for optimal assignment of vehicles to transportation tasks in terms of total cost and emission. The assignment problem is transformed into a process-network synthesis problem that can be algorithmically handled by the P-graph framework. In the proposed method, each task is given by a set of attributes to be taken account in the assignment; this is also the case for each vehicle. The overall mileage is calculated as the sum of the lengths of all the routes to be travelled during, before, after, and between the tasks (Desaulniers et al. 1998; Baita et al. 2000). Cost and emission are assigned to the mileages of each vehicle type. In addition to the globally optimal solution of the assignment problem, the P-graph framework provides the *n*-best suboptimal solutions that can be ranked according to multiple criteria. The viability of the proposed method is illustrated by an example.

## Keywords

P-graph Combinatorial optimization Vehicle assignment Transportation## List of Symbols

*T*Set of tasks

*S*Set of resources

*P*_{i}∈*T*Trip

*i*to be performed*t*_{s}(*P*_{i})Starting time of trip

*i**l*_{s}(*P*_{i})Starting location of trip

*i**t*_{e}(*P*_{i})Ending time of trip

*i**l*_{e}(*P*_{i})Ending location of trip

*i**d*Distance for each pair of locations

*R*_{k}∈*S*Vehicle

*k**l*_{a}(*R*_{k})Actual location of the vehicle

*k**c*_{t}(*R*_{k})The cost of vehicle

*k**e*_{t}(*R*_{k})The CO

_{2}emission of vehicle*k**v*_{max}(*R*_{k})The maximum speed of vehicle

*k**A*(*P*_{i})The set of resources potentially capable of performing task

*P*_{ i }*P*The set of the final targets to be achieved

*R*The set of the initially available resources

*M*The set of entities

*m*_{j}entity

*j**o*_{i}= (α_{i}, β_{i})Activity

*i*with*α*_{ i }set of preconditions and β_{ i }set of targets*O*The set of candidate activities

- \( L_{{p_{j} }} \)
Lower bound on the gross result

- \( U_{{p_{j} }} \)
Upper bound on the gross result

- \( U_{{c_{j} }} \)
Upper bound on gross utilization

*u*_{i}Upper bound for the volume of activity

*o*_{ i }*l*_{i}Lower bound for the volume of activity

*o*_{ i }- cm
_{j} Price for each resource on target

- cp
_{i} Proportional constant of activity

*i*- cf
_{i} Fixed charge of activity

*i**a*_{ji}The difference between the production and consumption rate of entity

*m*_{ j }by activity*o*_{ i }*m**Set of entities in the optimal structure

*o**Set of activities in the optimal structure

*x**The vector of the optimal volumes of activities

*z**Objective value of the optimal solution

## Notes

### Acknowledgments

Authors acknowledge the support of the Hungarian Research Fund under project OTKA 81493K.

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