Clean Technologies and Environmental Policy

, Volume 13, Issue 4, pp 637–642

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

• Mate Barany
• Botond Bertok
• Zoltan Kovacs
• Ferenc Friedler
• L. T. Fan
Original Paper

## 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

S

Set of resources

Pi ∈ T

Trip i to be performed

ts(Pi)

Starting time of trip i

ls(Pi)

Starting location of trip i

te(Pi)

Ending time of trip i

le(Pi)

Ending location of trip i

d

Distance for each pair of locations

Rk ∈ S

Vehicle k

la(Rk)

Actual location of the vehicle k

ct(Rk)

The cost of vehicle k

et(Rk)

The CO2 emission of vehicle k

vmax(Rk)

The maximum speed of vehicle k

A(Pi)

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

mj

entity j

oi = (αi, βi)

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

ui

Upper bound for the volume of activity o i

li

Lower bound for the volume of activity o i

cmj

Price for each resource on target

cpi

Proportional constant of activity i

cfi

Fixed charge of activity i

aji

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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## Authors and Affiliations

• Mate Barany
• 1
• Botond Bertok
• 1
• Zoltan Kovacs
• 1
• Ferenc Friedler
• 1
• L. T. Fan
• 2
1. 1.Department of Computer Science and Systems TechnologyUniversity of PannoniaVeszprémHungary
2. 2.Department of Chemical EngineeringKansas State UniversityManhattanUSA