Porting DICE to C/C++

We ported the GAMS version of the DICE 2013R model (DICE2013Rv2_102213_vanilla_v24b.gms) to C/C++ to more seamlessly integrate with the Borg-MOEA software. We tested the consistency of our port with the original GAMS code (Online Resource Fig. ESM1) and found that results from our port are virtually indistinguishable from those of the original GAMS code for the deterministic optimal solution with default parameters. The ported DICE model (CDICE2013) along with the coupled DOECLIM climate model (discussed below) are available at https://github.com/scrim-network/cdice_doeclim.

Representing key aspects of uncertainty in DICE

We represent a key uncertainty in DICE in order to calculate the expectations and reliabilities associated with each of the objectives. To do so, we produce 100 samples from a log-normal fit to a recently published distribution of climate sensitivity (Olson et al. 2012) (μ= 1.098001424, σ= 0.265206276) using inverse-transform sampling.

To improve the consistency of the model temperature hindcasts with historical temperature records, we couple the DOECLIM climate model (Kriegler 2005; Goes et al. 2011) to CDICE2013. DOECLIM is a simple energy-balance model that connects the troposphere and upper ocean with a diffusive deep-ocean layer. We calibrate the climate sensitivity, ocean diffusivity, and aerosol forcing parameters in DOECLIM using the National Aeronautics and Space Administration Goddard Institute for Space Studies (NASA-GISS) atmospheric temperature anomalies and radiative forcing data (Hansen et al. 2010) (Online Resource Fig. ESM2) from years 1900 through 2010. We calibrate the parameters by iterating over values of climate sensitivity while optimizing the ocean diffusivity and aerosol forcing parameters that minimizes the sum of square residuals between the model results and NASA-GISS temperature data (Online Resource Fig. ESM3). We acknowledge that using a Markov-Chain Monte Carlo analysis to obtain the full joint distributions of the three parameters would be a more formal and exhaustive calibration, but our approximate method seems sufficient for the purposes of this study.

A climate sensitivity sample with the calibrated ocean diffusivity and aerosol forcing parameters represent a single state of the world (SOW). For each SOW, a hindcast of atmospheric temperature anomalies is produced and used as the initial condition for the CDICE2013 projections (Online Resource Fig. ESM3). For every evaluation of DICE in the optimization process, the endogenous variables are recalculated for each state of the world, producing 100 values or time-series of each endogenous variable.

Coupling DOECLIM to CDICE2013 not only helps maintain the relationship among model parameters throughout the SOW sampling process, but shows that a better understanding of the science while using past (hindcast) information pushes the Pareto-optimal solution set closer to the ideal solution point (Online Resource Fig. ESM4). This improvement in objective space translates to possible additional satisficing solutions in decision space and thus an improved compromise across the preferences of the stakeholders.

Defining objectives

We define four objective functions.

$$ EUM={n_{sow}}^{-1}{\varSigma}_i\;{W}_i $$ (1)

$$ REL2C={n_{sow}}^{-1}{\varSigma}_i{\left\{{max}_t\left({T}_{AT,t}\right)\le 2.0\right\}}_i $$ (2)

$$ ABATE={n_{sow}}^{-1}{\varSigma}_i{\varSigma}_t\ {\varLambda}_{t,i}{\left(1+{ri}_{t,i}\right)}^{-t} $$ (3)

$$ DAM={n_{sow}}^{-1}{\varSigma}_i{\varSigma}_t\ {\varOmega}_{t,i}{\left(1+{ri}_{t,i}\right)}^{-t} $$ (4)

Where n SOW is the number of SOW and i is an index over each SOW. The EUM objective is similar to that defined in the original DICE model, except that we calculate the expectation of utility over the states of the world previously described. The REL2C objective is the fraction of SOW where the maximum deviation in atmospheric temperature (T AT ) at any time in the model projection t remains at or below 2.0 degrees Celsius. The expression max t (T AT,t ) ≤ 2.0 produces one if true or zero if false for SOW i. The ABATE and DAM objectives are the expectations over the SOW of the net-present value of the abatement costs (Λ) and climate damages (Ω), respectively, summed over time. The variable ri is the real interest rate calculated endogenously within the model and is a function of time t and SOW i. The EUM (Eq. 1) and REL2C (Eq. 2) objectives are maximized while the ABATE (Eq. 3) and DAM (Eq. 4) objectives are minimized.

Optimization

We use the Borg Multi-Objective Evolutionary Algorithm (Borg-MOEA) to search for the complex Pareto surface generated by the four-objective functions used in our problem formulation (Hadka and Reed 2012a). Borg-MOEA is an advanced evolutionary optimization algorithm and has been shown to be one of the most powerful optimization algorithms to date (Hadka and Reed 2012b; Woodruff et al. 2013). We chose Borg-MOEA as our optimization algorithm for its efficiency and reliability in finding the complex four-dimensional Pareto surface associated with our problem formulation.

The algorithm uses ε-dominance (Laumanns et al. 2002) as a means of numerical precision in objective space. We use epsilon values of 0.1, 0.01, 0.05, and 0.05 for the EUM, REL2C, ABATE, and DAM objectives respectively. This yields 2914 solutions (2251 solutions in the inertia constrained formulation) that create the four-dimensional Pareto-optimal surface.

We define convergence to the Pareto-optimal surface through the use of the Borg-MOEA operator selection probabilities and Pareto-improvements during the optimization (Online Resource Fig. ESM5, ESM6). Borg-MOEA uses a combination of operators that select the next potential solutions to be tested. Once the uniform mutation operator (UM) takes over the solution proposition process (i.e. high probability) and the Pareto-improvement rate drops to 1% or less of the size of the solution set, additional optimization time would yield only marginal gains in solution quality.