Lecture: Advanced Environmental Assessments

Transcription

1 Lecture: Advanced Environmental Assessments Carl Vadenbo

2 Agenda Introduction to optimization and operations research The general linear programming (LP) model The Simplex algorithm Why LCA operations research? Multi-objective optimization and Pareto-efficiency

3 What is optimization? an act, process, or methodology of making something (as a design, system, or decision) as fully perfect, functional, or effective as possible; specifically: the mathematical procedures (as finding the maximum of a function) involved in this [Assessed on ]

4 Operations Research (OR) Also referred to as decision science, management science, or operational research the application of scientific and especially mathematical methods to the study and analysis of problems involving complex systems Employing techniques from other mathematical sciences to arrive at optimal or near-optimal solutions to complex decision-making problems. Techniques include: mathematical modeling statistical analysis mathematical optimization [Assessed on ]

5 Origin of Operations Research George B. Dantzig Leonid Kantorovich

6 Example a cannery problem George B. Dantzig (1963) Linear Programming and Extensions. Princeton University Press, Princeton, NJ

7 The general linear programming (LP) model Objective function Subject to (constraints) min z = c T x or max! Ax b x 0 Variable (to be determined) Known coefficients

8 Illustration of LP formulation: the cannery problem Objective function min z = c T x Subject to (constraints) availability at canneries A 1 x b demand at warehouses A 2 x = d non-negativity of flows x 0 cost per case, c Chicago Dallas Kansas City New York San Francisco availability, b Portland $1.00 $1.80 $1.50 $0.90 $ San Diego $1.80 $1.60 $1.40 $2.50 $ Seattle $1.70 $2.00 $1.80 $2.50 $ demand, d

9 Graphical solution of LPs x2 max Z x

10 Solving LPs: the Simplex algorithm First proposed by Dantzig in 1947 If a LP has an optimal solution, then it must have an optimal solution that is also an extreme point of the feasible region Algorithm (conceptual description): 1. Identify a basic feasible solution If no such solution exist, then LP infeasible 2. If objective value can be further improved, then move along the edge that represents the largest (current) improvement If not, then STOP, the current solution is optimal If possible, then check if problem is unbounded (z ± ), if so then STOP, else proceed to Step 3 3. Derive new basic feasible solution and return to Step

11 cannery Chicago Dallas Kansas City New York San Francisco Back to the cannery problem the optimal solution Optimal solution: warehouse [Nr of cases] Supply Portland SanDiego Seattle Demand Objective value: $

12 Why LCA OR? Alternative description: OR deals with methods and models for supporting decisions arising in design, planning, coordination and/or control of (processes in) complex (socio-technical) systems Sounds familiar? Models of OR commonly focused on identifying optimal use of (constrained) resources, maximizing profit/minimizing cost LCA is (generally) based on linear homogenous [ ] models of human economic activity and of their effect on the environment (Azapagic & Clift 1998) Azapagic & Clift. (1998) Int. J. LCA 3(6):

13 Optimization with environmental objectives First applied to process design in chemical industry in the 90s, e.g. Methodology for Environmental Impact Minimization (MEIM) by Stefanis et al. (1995) Dissertation of Adisa Azapagic (1996) and publications ( ) LP to solve allocation problem of multi-output processes/systems LCA multi-objective LP Decision based on Pareto-efficient surface Current applications Process design and scheduling Planning of supply chain networks Maximizing benefits of by-product exchanges, i.e. industrial symbiosis Dealing with conflicting environmental objectives Stefanis et al. (1995) Computers & Chemical Engineering 19:S39 S44 Azapagic, A. (1996). Environmental System Analysis : The Application of Linear Programming to Life Cycle Assessment. PhD Dissertation, University of Surrey, UK.

14 Products, m Some LCA fundamentals the matrix-based LCI model As f Where A is the technology matrix (mxn), s is the scaling vector (nx1), and f is the final demand vector (mx1) Processes, n a a 11 1n m1 a a mn Three cases: Dimensions of m>n: More processes than products system overdetermined m=n: If each process has a unit product output, then technology matrix A is square As=f has a unique solution if A is invertible m<n: System underdetermined and can be optimized! Heijungs & Suh (2002), Tan et al. (2008) 14

15 Linear programming extension of matrix-based LCA model (LP-LCA) Key difference! For matrix inversion (standard LCA): technology matrix A must be square and invertible each process must have a unique product or service as output In LP-LCA: technology matrix A rectangular multiple processes compete to provide the same function or product (more columns than rows in A) Objective function 1 min h = gq s A f g h Bs gq Subject to (constraints) As = f Bs = g s

16 The curse of multiple objectives Issues Which performance indicators (objectives) are relevant? Computational effort How to visualize optimal solutions more than 3D? How to prioritize between conflicting objectives? Possible solutions Aggregation of objectives (e.g. single metric through weighting) Parameterization or ε-constraint method (Haimes et al. 1971) Pareto-optimal set of solutions Fuzzy LP formulation (Tan et al. 2008) Reduce complexity by omitting redundant objectives based on correlation (Guillén-Gosálbez 2011)

17 Pareto-efficiency/-optimality Buy a car f 1 (y i ) = Cost Parameters c i = cost of car i s i = safety level of car i f 2 (y i ) = -Safety Variables y i = binary variable (1 if car i is selected, 0 otherwise) Slides adapted from Gonzalo Guillén-Gosálbez 17

18 Cost Pareto-efficiency/-optimality Find a safer car for the same price Find a cheaper car with the same safety FEASIBLE SOLUTIONS (suboptimal: inefficient) UNFEASIBLE SOLUTIONS (impossible: not available in the market) PARETO FRONTIER (optimal: efficient solutions) Safety Slides adapted from Gonzalo Guillén-Gosálbez 18

19 Cost Pareto-efficiency/-optimality FEASIBLE SOLUTIONS (suboptimal) Reduce environmental impact X UNFEASIBLE SOLUTIONS (impossible) Reduce cost PARETO FRONTIER (optimal) Functional unit: key constraint! Environmental impact Guillén-Gosálbez et al Ind. & Eng. Chem. Research 47 (3)

20 The fuzzy LP extension of matrix-based LCA Objective function Subject to (constraints) max λ As = f Bs = g Qg h U λ(h U h L ) 0 λ 1 Degree of mutual satisfaction i.e. indicator of best compromise solution System variables Upper/lower bound for environmental target levels Where A: technology matrix; B: intervention matrix (env. flows); Q: characterization matrix of LCIA model; f: functional unit vector; s: scaling vector; g: inventory flows; h: impact vector in LCIA model Tan et al. (2008) J. Cleaner Prod. 16 (13):

21 Fuzzy LP formulation 1 acceptable partially acceptable unacceptable 1 acceptable λ partially acceptable unacceptable 1 acceptable λ partially acceptable 0 h Lower unacceptable h Upper impact i λ 0 h Lower h Upper impact 2 0 h Lower h Upper impact 1 Objective of fuzzy LP: Find maximum λ obtainable over all objectives simultaneously Tan et al. (2008) J. Cleaner Prod. 16 (13)

22 Image copyright under CC 3.0 from Case study: Environmental multi-objective optimization of biomass use for energy in Denmark

23 Goal & scope of study Aim: To determine (national) environmentally-optimal strategies for utilization of biomass resources for energy Goal: To overcome short-comings of scenario-based LCA Scope: Full set of biomass substrates, energy technologies, and final demand categories (complete energy system perspective) Approach: (i) consequential LCA of biomass supply and bioenergy conversion, (ii) biochemical process models, and (iii) mathematical optimization with multiple objectives

24 Biomass supply Problem superstructure Competing/compensatory energy technologies fossil + other renewables Primary conversions (e.g. anaerobic digestion) Secondary/ intermediate conversions (e.g. biogas upgrading) Tertiary conversions final service provision, (e.g. in CHP or for transport) Final energy demand 11 subcategories electricity heat transport

25 Image copyright under CC 3.0 from Case study scope Geographical scope: Denmark Departing from situation in 2013 (as baseline) Future energy scenarios 2025 frozen policy demand scenario Assuming renewable energy targets (as share of total) for electricity, heat & road/rail transport Six environmental objectives 4x ILCD-recommended midpoints (Global warming, marine eutroph., acidification, particulate matter Water footprint (WF), as water scarcity midpoint (Pfister et al. 2009) Cumulative energy demand (CED), non-renewable fossil resources

26 Single-objective optimal solution minimizing global warming (GW) Vadenbo et al. (in submission) Carl Vadenbo Energy System Analysis October

27 Multi-objective optimal solution Vadenbo et al. (in submission)

28 Discussion case study results Key case study conclusion: Context matters! Generally high degree of utilization for domestic biomass resources Dedicated energy crops in 3/6 single-objective solutions Bioenergy import: wood pellets/chips in 4/6, liquid biofuels in 4/6 (Too) high environmental cost of substituting animal feed Direct combustion and anaerobic digestion dominate 1 Bioenergy for heat and electricity prioritized over use as transport fuels Environmental win-win feasible Optimal solution robust for type of biomass utilized; less in terms of optimal conversion pathways

29 Wrap-up LCA combined with optimization Opportunities compared to scenario-based LCA Systematic identification of optimal solution Optimization problem can encompass very large number of feasible system configurations Optimization problem only considers feasible solutions Systematic identification of (Pareto-)efficient trade-offs between conflicting objectives Challenges compared to scenario-based LCA Formulation of relevant constraints (+data requirements) Computational effort Environmental optimization based on LCA consequential LCA!

30 Introduction to Environmental Computer Lab (ECL I) on optimization :00 a.m. in HIL E 15.2 on Tuesday December 20 th Aim of exercise Introduction to LP solver in Excel To be able to formulate and solve a small LP problem (Introduction to shadow or marginal costs) Desktop computers available, own laptops also ok

31 Links Introduction to Mathematical Optimization by IFOR (fall semester) /introduction-to-mathematical-optimization.html Introduction to LP and its implementation in Matlab: Linear Programming and Extensions by George B. Dantzig: An introduction to linear programming by Tom Ferguson (UCLA):