1 / 21 A mass-fow MILP formulation for energy-efficient supplying in assembly lines Maria Muguerza Cyril Briand Nicolas Jozefowiez Sandra U. Ngueveu Victoria Rodríguez Matias Urenda Moris LAAS-CNRS, ROC team (Operations Research, Combinatorial Optimization, Constraints), Toulouse MISTA, Prague, August 25-27 2015
2 / 21 Outline 1 Introduction 2 Problem definition 3 Energy analysis 4 Mathematical modeling 5 Experimentation 6 Conclusion
3 / 21 Introduction ECO-INNOVERA Project - EASY (Energy Aware Feeding Systems) Skövde University / Volvo VCE Universidad de Navarra / VW Polo LAAS-CNRS, Toulouse Motivations Develop more sustainable internal logistic practices Improve the energy efficiency of production systems Focus on the supplying system of assembly lines
Introduction Problem definition Energy analysis Mathematical modeling Experimentation Conclusion Introduction EASY s objectives Energy oriented analysis of supplying systems Determination of the most significant energy parameters Design new effective optimization and simulation methods Help decision-makers to find a good balance between energy and economical efficiency - Energy awareness 4 / 21
5 / 21 Outline 1 Introduction 2 Problem definition 3 Energy analysis 4 Mathematical modeling 5 Experimentation 6 Conclusion
standardized bins with a specific capacity for each part. Introduction Problem definition Energy analysis Mathematical modeling Experimentation Conclusion The capacity limit for each wagon of a tow train is almost 50 bins, and no more than two wagons are allowed per tow train. Furthermore, each assembly line is supported by a single supermarket, which makes it easier and more practical by avoiding any interruption since Problem all the requests description and deliveries are managed at a single place near the assembly line. In addition, the assembly lines are two-sided where both sides of the line (left and right) are used in parallel. A schematic view of the assembly line with a single supermarket is Assembly represented in line Figure supplying 1. Route Workpiece Rack Rack Supermarket Tow train Figure 1. A straight assembly line view with the supermarket-concept Assumptions 4- Description Periodic of the model supplying A single vehicle / A single route In this section, an optimization model, which is a mixed integer linear programing (MILP) Decision variables model, is presented based on the VW-Navarra feeding process. The MILP model is a model for optimally Determine loading tow which trains and workstations scheduling the todeliveries. serve atthe each improvement period criteria in this model Determine are the number theof number tours and ofthe components inventory level. to be It is delivered worth noting that the improvement Constraints: criteria and their priority are selected according to our experience at VW- Navarra. The Avoid indices, component parameters and shortage decision / variables satisfy component for the MILP model demands are presented along in Table1. periods Respect each workstation capacity / vehicle capacity Table 1. Notation for the MILP model 6 / 21
7 / 21 Performance objectives Comparison Energy minimization (vehicle) vs. Distance (Tours) minimization Energy minimization Pollution Routing problem - Betkas and Laporte, Transportation Res. Part B, 2011 Energy Minimizing VRP - Kara et al.,verlag Berlin Heidelberg 1, 2007. Pollution Inventory Routing problem - Shamsi et al., Logistic Operations, 2014 Survey of Green Vehicle Routing Problem - Lin et al. In: Expert Systems with Applications, 2014
8 / 21 Outline 1 Introduction 2 Problem definition 3 Energy analysis 4 Mathematical modeling 5 Experimentation 6 Conclusion
9 / 21 Energy analysis A bit of physics E = P dt P = F v More relevant forces: Traction force (F t ) F t = m T a(t) Rolling resistance (F r ) F r = m T gc r Energy equation: E = m T (a(t) + gc r ) v(t)dt
W Energy consumption profile Assumption on the acceleration profile Power Vs Time (between 2 stations) 3000 1 2 3 2000 1000 0 s 0 5 10 15 20 25 30 35 40 Properties Any vehicle stop produces a pic of energy consumption. The consumed energy is proportional to the carried mass. 10 / 21
11 / 21 Outline 1 Introduction 2 Problem definition 3 Energy analysis 4 Mathematical modeling 5 Experimentation 6 Conclusion
Model basis A transportation network A node (i, t) in the network = a workstation i at period t A mass flow circulates from the depot along the network at each period t (plain arcs) A energy cost c i,j is associated to each arc A component flow circulates from workstation i at period t to workstation i ar period t + 1 (dotted arcs) I t 1 i M t j,i with j<i c j,i i,t c i,j M t i,j with j>i I t D t 12 / 21
13 / 21 Model basis A 2-periods / 4-workstations network 1,1 2,1 3,1 4,1 0 1,2 2,2 3,2 4,2
A MILP formulation Decision variables M t ij : mass going from i to j during period t Z t i : components delivered to workstation i at period t IL t i : inventory level of workstation i at period t φ t ij = 1 whether the vehicle is going from i to j at period t Constraints Objective Mass, component, vehicle flows conservation Vehicle / stock capacity Demand satisfaction Energy minimization : n NT i,j t c ij Mij t NP-hard in the strong sense... 14 / 21
15 / 21 Min z = NT c ij Mij t + NT c ij m v φ t ij (1) i,j t i,j t st: Z t i + IL t 1 i Z t i 1 ( m i j<i d t i = IL t i (i, t) (2) M t ji M t ij ) = 0 (i, t) (3) j>i Z t i + IL t 1 c i i (i, t) (4) φ t ij = φ t ji (i, t) (5) j>i j<i M t ij m maxφ t ij (i, j, t) (6) IL t i 0 (i, t) (7) M t ij 0 (i, j, t) (8) Z t i N (i, t) (9) φ t ij {0, 1} (i, t) (10)
16 / 21 Outline 1 Introduction 2 Problem definition 3 Energy analysis 4 Mathematical modeling 5 Experimentation 6 Conclusion
17 / 21 Experimentation Problem instances Instances adapted from those (IRP) of C. Archetti et al., Transportation Science (2007). Each workstation demand is periodic and assumed constant One single route among the customers is imposed New added parameters: energy cost, vehicle mass, component mass, maximum vehicle load 9 instances for each NT = {5, 10, 20} and N = {5, 15, 30, 50} values were created 108 instances Optimization strategy First, distance (tours) minimization (GUROBI) Then energy minimization (GUROBI, Time Limit = 300sec) Comparison of the energy consumed by both solutions
18 / 21 Results Time performance (sec) 140 120 100 80 60 40 20 Time performances (sec) 0 0 5 15 30 50 N (worksta+ons) T=5 - Tours T=5 - Energy T=10 - Tours T=10 - Energy T=20 - Tours T=20 - Energy For NT = 20 and N = 50, 2 instances were not solved to optimality (GAP < 1%)
19 / 21 Results Energy savings (%) 8,00% 7,00% 6,00% 5,00% 4,00% 3,00% 2,00% 1,00% 0,00% Energy profit (%) 5 15 30 50 N (worksta+ons) T=5 T=10 T=20 For every instance, the optimal number of tour remains unchanged when energy is minimized.
20 / 21 Outline 1 Introduction 2 Problem definition 3 Energy analysis 4 Mathematical modeling 5 Experimentation 6 Conclusion
21 / 21 Conclusion and perspectives Conclusion Significant energy parameters : distances, masses, number of stops. An effective MILP model for energy minimization Energy optimization does not cause the increase of the number of tours Perspectives Improvement of the problem instances Need of more advanced optimization techniques Generalization to the Inventory Routing Problem