Multi-criteria comparison tools to evaluate cost- and eco-efficiency of ultra-high-performance concrete

This work was motivated by the increasing need for proper metrics and tools to demonstrate the effect of mechanical performance, as a function of concrete mix composition, in dictating the dimensions of structural elements and associated costs and embodied carbon dioxide (CO2) emissions. Mixture compositions associated with different concrete technologies were compared using multi-criteria comparison indices derived using structural design considerations and calculated using information on compressive strength, volumetric embodied CO2 and unit costs. In addition, predicted compressive strengths obtained with machine learning (ML) models are used to calculate these indices for a domain of mix proportions associated with ultra-high-performance concrete materials to generate multi-objective density diagrams (MODDs). The makeup of this tool facilitates the evaluation of rather complicated trends associated with mix proportions and multi-objective outcomes, allowing ML-based tools to be of easy interpretation by industry personnel with no expertise in artificial intelligence. MODDs could be used as aids in the decision-making process during mix design stages and serve as proof of mixture optimization that could be introduced in environmental product declarations. Results show that, in contrast to conventional wisdom, high-binder content and ultra-high strength concrete technologies are not necessarily detrimental to cost and/or eco efficiencies. For the applications evaluated herein, optimum solutions were mostly obtained with these types of concrete, suggesting that industry trends toward requiring minimization of embodied carbon footprint on a per volume of concrete basis are misguided and should not be used as a standalone metric to minimize the total carbon footprint of concrete structures.


Introduction
According to the American Concrete Institute (ACI), the definition of ultra-high-performance concrete (UHPC) is reserved for concrete materials with compressive strengths over 150 MPa 3 (22 000 psi), with specified tensile ductility, durability, and toughness requirements [1]. The advanced properties of UHPC stimulate the development of concrete elements with more efficient cross-sectional areas, typically resulting in less volume of material required [1] compared to other traditional concretes, and less associated maintenance and repair costs due to its exceptional durability [2]. Additionally, the presence of steel fibers allow partial (or at times, full) replacement of transverse steel reinforcement [3]. GHG emissions [11]. Research is being conducted to improve the eco-efficiency of cement manufacturing, either through electrifying the heating process [24] or through the electrochemical synthesis of lime during the calcination process [25], which is reportedly allowing 75% reductions in carbon emissions. Yet, the scalability of these processes remains a question mark, especially if it requires replacing major equipment in existing cement plants. With global population growth and fast urbanization of developing countries, it is unlikely that these technologies can be promptly implemented to meet environmental needs considering that material demand is expected to double by 2050 [8]. Given these time constraints, international leading entities have recommended further evaluation of material property changes associated with constituent alteration as a timely-convenient strategy to minimize GHG emissions [26].
Currently, EPD has been implemented as a product 'label' to promote transparency with respect to efforts made towards improving the sustainability of concrete mix design. Usually, EPDs quantify a wide range of environmental impacts (e.g. GHG, toxic substances, water impacts, ozone depletion, habitat destruction, etc) throughout the product's life cycle (cradle to grave) [27,28]. The effect of GHG is typically summarized by the 'CO 2 -eq' term, which describes their GWP as defined in the IPCC Fifth Assessment Report [29].
To facilitate the calculation of volumetric environmental impacts associated with concrete production, several life-cycle assessment (LCA) tools have been developed throughout the years [30][31][32][33][34][35][36][37][38]. Yet, there is still no consistent analysis or reporting method used at a global scale to quantify and compare emissions generated in construction based on constituent proportioning in mixtures and the resulting performance of the final product. While EPDs currently report GWP for concrete in units of kg.CO 2 -eq m −3 , this indicator does not promote a consistent and accurate quantification of environmental impacts for the totality of a project. For instance, normal-strength concretes (NSC) often require larger structural elements to withstand the same design loads compared to UHPC. This trade-off clearly influences the total amount of GHG emissions considering the savings in volume of material obtained when using UHPC. In many applications, using UHPC could well represent a more eco-efficient solution compared to other traditional concretes, with less GHG emissions associated with casting the entirety of the given element. Yet, metrics and tools that clearly demonstrate this reality are still not well developed. Until addressed, this knowledge gap could continue to affect the industry's perception of high-binder-content materials such as UHPC.
Many researchers have investigated sustainable and economical approaches to design UHPC materials by focusing on reducing the contents of cement and silica fume [39][40][41]. While many strategies for reducing carbon footprint are readily implementable [42], most studies have focused on evaluating the environmental impact associated with manufacturing raw constituents [43][44][45][46]. Lately, emphasis has been placed on alternative cements such as calcined clay limestone (LC 3 ) [47] or Portland limestone cements [48]. Nevertheless, the path for sustainable construction is contingent on the many parts involved in projects (architects, structural designers, concrete producers, etc). While alternative cements can greatly contribute to reducing embodied CO 2 from concrete mixtures, the influence of mechanical performance in dictating design is still under-investigated and not fully understood.
Recently, significant efforts towards accurate quantification of concrete's environmental impacts have been undertaken by Miller et al [19,20], who proposed pilot comparison indices for reinforced and unreinforced concrete members, and later by the authors [21], who proposed further modifications to expand the applicability of one particular index to concretes with all ranges of compressive strengths.

Cost-and eco-efficiency comparison indices
Metrics that account for the role of mechanical properties in determining the volume of structural elements were first proposed by Miller et al [20] for unreinforced concrete members. These indices were established from environmental impacts associated with cradle-to-gate production of one unit volume of concrete, in addition to mixture parameters and material properties. These metrics were based on simplified cases, where only one material property was defined as the controlling factor for performance. Later on, Kourehpaz and Miller [19] proposed indices to account for steel reinforcement in axially loaded short columns and simply supported beams under uniform distributed loads.
To illustrate the influence of mechanical performance and volumetric environmental impact on the design of structural members, an index was derived in [21] for simply supported beams under uniform distributed loads. For this case, load, width, and length were assumed as specified by design. On the other hand, beam depth was defined as a free variable, modeled as a function of the required moment of inertia as determined by the strength of the concrete. For this scenario and assuming a rectangular cross-section, the maximum moment in a simply supported beam subjected to a uniformly distributed load was compared against the moment required for a crack to initiate in the outer fiber of the concrete. Arranging these equations as a function of depth h results in where b is the width of the member, w is the distributed load, l is the length of the member and f r is the modulus of rupture. Meanwhile, the total environmental impact I associated with a given member can be written as where A s is the cross-sectional area of steel, i c is the volumetric environmental impact of concrete and i s is the volumetric environmental impact of steel. To incorporate the compressive strength ( f c ) of a given concrete mixture into this assessment, f r in (1) was estimated from f c based on different ACI equations ( [1,49,50]), which varies with the type of concrete. A coefficient, λ fr , was derived in [21] as with f c in units of MPa, to relate f c and f r . Despite not being explicitly stated in [21], f c and f r are related through λ fr as Substituting equations (1) and (4) inside equation (2), the total environmental impact I of the reinforced concrete beam, designed for initial cracking under the conditions aforementioned, can be written as As discussed in [21], an index can be derived for this member to evaluate the contribution of the concrete to the environmental impacts. For applications where the area and strength of steel are held constant, and b, w and l are imposed by design, an index that enables comparison between different concrete mixtures based on initial cracking due to bending was proposed by the authors in [21] as When evaluating this metric, the objective consists in minimizing this term to reduce environmental impact associated with the designed member. Other indices were also developed by Miller et al for the yield and nominal stages of bending [19], as well as other controlling material properties affecting durability such as fracture toughness, chloride diffusion and thermal conductivity [20]. To maintain focus on the tools developed in this study, these indices will be addressed in future work.
An additional index proposed by Kourehpaz and Miller [19] is used herein to compare the axial capacity of short columns when the effects of buckling are negligible. To derive this index, the design load (or axial force expected to be resisted by the reinforced concrete column) F column can be written as Arranging equation (7) into equation (2), the total environmental impact for the column can be written as To compare the environmental impacts of different concrete mixtures in scenarios where the area and strength of steel rebar are fairly constant, an eco-efficiency index can be defined as To simplify cost analysis, equations (6) and (9) were modified in [21] to define cost-efficiency indices as and where u c is the mixture cost per unit volume of concrete, ρ column is the index used for short columns and ρ cracking is the index associated with the initial cracking of simply supported and uniformly loaded beams subjected to bending. The volumetric cost, u c , of each mixture was calculated as a product of the quantities used for each ingredient per cubic meter of concrete and their unit price. The costs associated with batching, transportation and placement were assumed to be the same for all mixtures 4 .

Multi-objective optimization
The process of methodically and concurrently optimizing a set of objective functions is known as multi-objective optimization or vector optimization. Different approaches can be followed in these types of problems to model a decision maker's preference depending on how the user articulates these preferences. 'Prior articulation of preferences' implies that the relative importance of each objective function is determined by the decision-maker prior to the optimization process. On the other hand, a 'posteriori articulation of preferences' involves selecting a particular solution from a set of mathematically equivalent solutions. Methods that require no articulation of preferences have also been investigated [51,52]. Other function transformation methods can be followed [53][54][55][56] to improve robustness of the optimization process. Also, different methods can be followed for multi-objective optimization problems involving competing objective functions. For instance, if a problem consists in minimizing one outcome (e.g. cost) while maximizing another outcome (e.g. flowability), methods such as grey relational analysis [57] can be used to normalize these outcomes to factors generated with different objectives (lower-the-better vs higher-the-better) and use them concurrently in one single objective function. Other approaches can also be followed for the multi-objective function formulation to address problems where the weighted summation is unable to capture optimum points. Example are the ε-constraint [58], weighted min-max method [59], exponential weighted criterion [60] and the weighted product method [61]. Further discussion on multi-objective methods can be found in the literature [62], including important concepts such as Pareto optimality [63]. The multi-objective formulation herein follows a classic weighted summation of each objective function (cost-and eco-efficiency indices), which are normalized by the worst solutions available in their corresponding domains. This simplified approach was followed considering that both cost and emissions are objective functions for which optimum solutions are associated with minimizing their outcomes.

Concrete mix design optimization: new opportunities with AI
The quality and proportioning of ingredients are major factors dictating mechanical performance, cost and environmental impact of concrete materials. Considering that most UHPC compositions [1,[12][13][14][15] require high cement dosages and increased costs per unit volume, it is imperative to develop smart mix design strategies to design structural UHPC elements that can compete financially and environmentally with other traditional concrete materials.
Throughout the years, several authors have investigated UHPC mix proportioning optimization [14,15,41,64]. This involves satisfying multi-criteria performance levels (different materials properties, carbon footprint and cost), which is challenging due to the synergistic relationship between the various constituents and the various desired outcomes. This synergy makes linear regression ineffective in modeling such complex domains. To address this, extensive research has been done on non-linear regression and computational methods that leverage the multitude of experimental data concerning advanced concrete materials, advanced mathematical techniques, and the power of high-performance computing [65][66][67][68]. Meanwhile, AI techniques, especially ML, have gained momentum in optimization and prediction studies [69][70][71]. A variety of algorithms can be used to accomplish the same goal: detect patterns in datasets and improve predictive performance.
Despite the growing popularity of ML in civil engineering, the 'black-box' nature of these algorithms tends to complicate the interpretation of the predictive structure of generated models. These often contain complex internal structures, described by parameters that offer little insight on what takes place inside the model during computations. Nevertheless, many researchers have used AI for concrete mix design optimization [72][73][74][75][76][77], while others have focused on fractional factorial design techniques to reduce experimental runs [78,79]. Recently, these two approaches were combined by the author in [22,80] to effectively estimate the trends existing in the evaluated domain of mix proportions without resorting to extensive experimental campaigns or use multiple-source datasets. In particular, orthogonal arrays were combined with ML techniques to develop performance density diagrams (PDDs). This mix design tool resembles a matrix of contour plots, allowing mixture proportioning and performance indicators to be visualized concurrently, while strategically using data from reduced experimental runs to develop the predictive structure of the model. The synergistic effect of multiple features on a given outcome can be evaluated concurrently using PDDs with multiple faceting strategies. Figure 2 (from [22]) illustrates the typical structure of PDDs, where the magnitude of the outcome (z axis) is displayed in the diagram as the density factor while the most important variables are placed in the x and y axes and the least important variables are discretized by blocks.
This methodology will be extended in this study to generate multi-objective diagrams associated with UHPC mix compositions. It should be noted that these diagrams can be developed for any class of concrete, using predicted values generated by any available model (e.g. [72][73][74]77]) provided that the model used is robust and capable of effectively predicting the trends within the domain of the experimental dataset.

Compressive strengths used to calculate the proposed comparison indices
The compressive strength results (f c ) used to generate the diagrams in section 5.1 correspond to predictions obtained by the authors in [22] when modeling compressive strength as a function of mix proportions. The design of experiments consisted of orthogonal arrays while the data was modeled using an ensemble technique that combines two ML models: k-nearest neighbors and random forest. All computations were performed in the data analysis software 'R' using the built-in algorithms from the caret package [81]. The domain in question corresponds to UHPC mixtures with a quaternary binder blend with fixed proportions by weight (by wt.): 68.3% cement, 22.4% slag, 5.3% microsilica and 2% fly ash; and varying aggregate contents. The aggregates used consisted of a natural silica sand (denoted as concrete sand), a crushed sand and a ground quartz filler. The maximum content for each type of aggregate was fixed at 25% by wt. of cementitious binder replaced in the system. This work focused on using these predicted compressive strengths to calculate multi-criteria comparison indices affecting environmental impact and cost for the evaluated domain of mix proportions. Detailed information related to defined boundaries, raw ingredients, mixing, curing and testing procedures can be found in [22]. Figure A1 in appendix A illustrates a PDD describing the predicted compressive strengths for the domain of mixture proportions investigated in [21]. Data analysis from that study indicate that the generated models and PDDs effectively predicted the trends, magnitude and ranking of the mixtures tested experimentally and stored in a test set, providing validation to the proposed method. Later in section 5.2, optimum mixtures identified by the models in [22] and tested experimentally are compared against different concrete mixtures from the literature, using the multi-criteria comparison indices proposed in this study.

Environmental impact factors and unit costs of raw ingredients
Several LCA tools have been developed in the last few years [30][31][32][33][34][35][36][37][38] to calculate the environmental impacts associated with concrete manufacture. Given that LCA analyses are out of the scope of this work, the environmental impact factors used herein were derived from a study by Celik et al [43]. In that study, a cradle-to-gate LCA approach was used to assess direct and supply-chain GWP associated with materials used in concrete production. The boundary of concrete production system incorporates: extraction and processing of aggregates, superplasticizer production, preparation and treatment of fly ash prior to mixing into concrete, extraction and processing of limestone, and concrete batching, and transportation of raw materials and products within the system. Table 1 illustrates how the volumetric GWP (kg CO 2 -eq m −3 of concrete) of each ingredient was used to derive the GWP factor on a weight basis (kg CO 2 -eq kg −1 ) required to assess the embodied CO 2 for the mixtures evaluated in the present study. Additional environmental impact factors, obtained from sources other than [43], are referenced in the table.
The GWP associated with slag and microsilica were assumed equivalent to the one estimated for fly ash in Celik et al [43]. To estimate the GWP of the crushed sand and ground quartz aggregates investigated herein, the GWP from coarse aggregate production in Celik et al [43] was combined with aggregate crushing factors estimated from Santero et al [82] to differentiate the environmental impact from these aggregates. Moreover, in Celik et al [43], GWP associated with raw material transportation can be mostly attributed to fly ash, water-reducer admixture and coarse aggregate. This is due to the considerably longer distances required to transport these materials in comparison to the others. However, for the mix design used to derive the impact factors herein, the aggregate-to-cementitious ratio was 4:1, while 30% (by weight) of cement content was replaced with fly ash. Thus, for each unit volume of concrete produced, the ratio of coarse aggregate to fly ash transported is over 13. This ratio is even higher when comparing aggregates and water-reducer admixtures. Therefore, given that transportation is the main source of CO 2 emissions associated with aggregate use, the authors allocated the entire transportation value to the aggregates in the study herein.
The environmental impact factor for steel fibers was estimated from an EPD in [90]. Unit costs from each ingredient were estimated based on various sources [83, 84, 86-89].

Proposed multi-criteria comparison indices
Carbon penalties and incentives are being pushed by governments and policy makers to help meet environmental goals [91][92][93]. From the concrete producer's perspective, it is important to meet performance and sustainability requirements while maintaining a cost-efficient business. Therefore, design efficiency should ultimately be measured using comparison indices that account for cost and emissions concurrently. In this work, a multi-objective comparison index for a short column MO column is defined as whereρ column is the column cost-efficiency index ρ column normalized by the worst solution (ρ max ) within the experimental domain,χ column is the column eco-efficiency index χ column normalized by the worst solution (χ max ) within the experimental domain, and α and β are the weighting coefficients. The approach used to normalize the indices ensures that MO column varies between zero and one, facilitating comparisons involving indices with different magnitudes. The same logic can be applied to calculate MO cracking . It is tempting to predefine the weighting coefficients by allocating the same weight to cost and environmental efficiencies (α = 0.5; β = 0.5). As it will be demonstrated in the subsequent section, care should be taken when allocating these weights considering that small savings in material cost can be accompanied by enormous environmental implications and vice-versa. It should be obvious that mix designs selected for a given project are rarely optimum for all the different structural members. Yet, implementing different concrete grades in the same structure is challenging and highly dependent on the practicality of suppling operations, as well as local regulations and codes. To account for applications where one type of concrete is used for the entire infrastructure, multi-member indices must be considered. Thus, multi-member eco-efficiency,MM emissions , and multi-member cost-efficiency, MM cost , indices could be defined, respectively, as MM emissions =v c .χ column +v b .χ cracking (13) and MM cost =v c .ρ column +v b .ρ cracking (14) where thev c andv b coefficients represent, respectively, the estimated volume percentage of columns and beams relative to the entire project. It is important to note that the results and trends observed in the following section are a function of the GWP and unit cost factors assumed to calculate the proposed multi-criteria indices. Changes in material costs (with different suppliers) and/or GWP factors (with source, region, or even LCA tools used) can significantly impact the results. Therefore, generalization of the observed trends should be avoided.

Multi-objective density diagrams (MODDs) for sustainable, cost-effective solutions
As previously mentioned, the database of mix compositions and their corresponding compressive strengths used herein was obtained from [22] (in particular from 'Phase B') to calculate the proposed multi-criteria indices. This domain consists of UHPC mixtures with varying replacement levels (up to 25% by wt. of cementitious) for three different aggregates (concrete sand, crushed sand and ground quartz), a fixed binder system (68.3% cement, 22.4% slag, 5.3% microsilica and 2% fly ash by wt.) and a fixed water-tocementitious ratio (w/cm = 0.20). For the short column case, using table 1 to calculate the volumetric GWP and cost of each mixture, the MO column index calculated for this domain can be described by the MODD illustrated in figure 3.
This index was calculated assuming equal weight for cost and eco-efficiency (α = 0.5; β = 0.5). The challenge with using MODDs as presented in figure 3 is that, when selecting an optimum mix based solely on minimizing MO column , the mechanical performance is not intuitively determined. Deriving f c from the value of the index requires the estimated volumetric GWP of the corresponding mix. This iteration can become cumbersome if optimum mixtures fail to meet performance needs. To account for mechanical performance thresholds determined by structural design, these diagrams can be built while filtering mixtures below a defined limit. For instance, if a given application requires an f c over 100 MPa, a filtered version of figure 3 can be plotted as shown in figure 4. This diagram shows several optimum alternatives to obtain a combined cost and eco-efficient solution while still satisfying the performance demands imposed by a priori structural design.
Observing figure 4, examples of optimum mixtures for this application could be: (1) 20% concrete sand, 8% crushed sand and 10% ground quartz; or (2) 25% concrete sand, 8% crushed sand and 12.5% ground quartz; or (3) 15% concrete sand, 25% crushed sand and 17.5% ground quartz. This tool allows designers and producers to make proper considerations regarding material availability, maximizing the use of local resources. Similar analysis can be done for the multi-member indices suggested in section 4. Appendix C contains tables illustrating the results for these indices, suggesting that optimum concrete mixtures for a given member may differ from the optimum obtained for a different element (e.g. a beam versus a column).  For small scale projects, combined multi-member, multi-objective indices could be derived to evaluate the best mix design solution for the whole structure. The weighting coefficients allocated to cost and eco-efficiency greatly impact the determination of optimum solutions. For developing policies and regulations, it would be tempting to pre-establish these weights. This could be done either by allocating the same weight to cost and environmental efficiencies, or by shifting most of the weight to the environmental index in an effort to pursue top eco-friendly solutions. However, there are advantages to viewing the optimization problem more holistically by carefully evaluating the tradeoffs when optimizing a mixture for a given application. Table 2 provides data to support this argument. For the short column case, while shifting most of the weight to the environmental index (β > 0.5) can result in the most eco-efficient solution, it also results in low rated mixtures in terms of cost-efficiency. For instance, the mixture corresponding to β > 0.5 (MO-Col-6) is under the top 50th percentile in cost-efficient solutions. Meanwhile, allocating equal weights to α and β helps identifying a more balanced mixture (MO-Col-5), which is still in the top 6th percentile for eco-efficiency, though much improved in terms of cost-efficiency (top 42nd percentile). Setting α at 0.6 and β at 0.4 provides a mix (MO-Col-4) that is also worth discussing, sitting in the top 27th percentile in terms of eco-efficiency and top 13th percentile for cost-efficiency.
To understand the magnitude of the impact associated with changing these coefficients on the total cost and carbon footprint of these columns, a case study is presented herein. For this example, columns are assumed to be primarily loaded in axial compression, assuming negligible moments at the column ends (plausible for interior columns under gravity loading). The axial load resisted by the column (design load) is specified at 22MN, representative of an axial load demand in a typical 15-20 story building structure. A length of 3 m is imposed by design. Table 3 describes the mixtures evaluated in this discussion.
To calculate the total environmental impact associated with designing a column with each mixture listed in table 3, equation (8) from section 2.2 is used. To maintain the focus of this work, equation (8) is calculated considering the contributions of concrete alone 5 . Similarly, the total cost is calculated using equation (8) by substituting i c with u c . The analysis performed herein extends the calculations to a total of 300 columns to simulate the impacts at the scale of a typical 15-to 20-story building with 20 columns per floor [94]. Figure 5 summarizes the results.
Observing figure 5(a), it is evident that the mixture that produces the column with the lowest embodied CO 2 , MO-Col-6, does not coincide with the mixture with the lowest volumetric CO 2 , X-Col-2. Casting a column with mixture MO-Col-6 (for which i c = 624 kg.CO 2 -eq m −3 , f c = 95 MPa) results in about 434 kg.CO 2 -eq in emissions whereas mixture X-Col-2 (i c = 552 kg.CO 2 -eq m −3 , f c = 82 MPa) results in about 444 kg.CO 2 -eq in emissions. This is explained by the differences in the compressive strength, which results in different cross-sectional areas and consequently different volumes of material required for each mix. In a building with 300 columns, that is 3 tons.CO 2 -eq in excess (see figure 5(d)). The same observation can be made in terms of cost, where using the mixture with the lowest volumetric unit cost, p-Col-2, does not result in minimizing the total cost of the column. In fact, two other mixtures, MO-Col-1 and MO-Col-4, result in cheaper columns (see figure 5(a)). The significance of using these indices for mix design selection is evident when comparing the trade-offs between using the most eco-efficient mixture (MO-Col-6) against the most cost-efficient mixture (MO-Col-1) for an entire building (contribution of the 300 columns). As it can be observed in figure 5(d), selecting MO-Col-1 over MO-Col-6 allows saving approximately $12 700 in total cost of columns. However, MO-Col-1 produces nearly 100 tons of CO 2 more than MO-Col-6 when casting a column under these conditions. This shows how mixture design optimizations that only consider costs can have tremendous environmental consequences with no considerable financial gains. Observing figure 5(d), mixtures MO-Col-6 and MO-Col-5 appear to be the most optimum mixtures to attain a balanced tradeoff between costs and carbon footprint. Similar discussions can be held to find optimum solutions for MO cracking using table B1 in appendix B. Similarly, tables C1 and C2 in appendix C could be used to extend this discussion to the multi-member indices.

Comparison between different concrete technologies using the proposed comparison indices
To extend the discussion to different concrete technologies, the proposed efficiency indices are used to compare mixtures from different types of concretes. The data were collected from various studies from the literature [20,41,85,[95][96][97][98][99], including results from [21], and mixtures defined in tables 2, B1, C1 and C2. Multi-objective indices were calculated using equal weighting coefficients (α = 0.5; β = 0.5). Supplementary materials [100] contain detailed information regarding source and properties for each mixture discussed in the following plots. Mixtures from the literature were tested at ages 28 and 90 d whereas mixtures from [21] were tested at age 56 d. To enable proper comparisons, strengths from the literature were averaged between the two testing periods to obtain an estimated 56th day strength. Figure 6 illustrates the cost-and eco-efficiency indices as a function of unit cost (u c ) and volumetric environmental impact (i c ), respectively. As previously discussed, the best suited mixtures for each index/application displayed correspond to the ones that minimize these efficiency indices.
Observing figure 6 and the plots associated with short column indices (χ column ,ρ column and MO column ), it is evident that there is very little correlation between volumetric impact indicators (i c and u c ) and the efficiency indices. Meanwhile, observing the plots associated with initial cracking indices (χ cracking , ρ cracking and MO cracking ), the correlation is stronger between volumetric impact indicators and efficiency indices for initial cracking applications. This is mostly due to the reduced influence of compressive strength on equation (11), considering that f c is to the power of 1 /4. Yet, there is a clear shift between the curves with changes in material type, which is associated with increased compressive strengths. This is due to the difference in the modulus of rupture between these different types of concrete, which was accounted for using λ fr . Therefore, cost and eco-efficiencies between different types of concrete should not be compared based on volumetric indicators. Evidence of this is given by observing figure 6 and the plots associated with initial cracking applications, where a UHPC mixture with approximately 900 kg CO 2 -eq m −3 appears to be as eco-efficient as an high-strength concrete (HSC) with 500 kg CO 2 -eq m −3 or a NSC with 300 kg CO 2 -eq m −3 . The same analysis can be done in terms of material costs, where a UHPC mixture costing approximately $330 m −3 is as cost-efficient as an HSC costing around $180 m −3 (these values do not include the cost of fibers). To help address the myths related to the environmental impact and cost-efficiency of UHPC materials, these efficiency indicators (including the multi-objective indices) are evaluated as a function of compressive strengths and binder content in figures 7 and 8, respectively. Figures 7 and 8 suggest that high paste content, high strength (and ultra-high strength) concrete technologies are not necessarily detrimental to cost or eco-efficiencies and these are not a function of the type of concrete applied. For the different indices evaluated, nearly equivalent optimum solutions can be obtained with almost all types of concrete. In particular, for the short column application, HSC and high-performance concrete (HPC) mixtures (green diamonds) and UHPC mixtures (blue triangles) appear to be the most efficient solutions. Observing figure 8, it is visible that despite being correlated to the volumetric impact indicators i c and u c , binder content has no clear correlation with any of the efficiency indices. On the other hand, figure 7 indicates that very few UHPC mixtures are cost-effective for the initial cracking application ρ cracking when compared to the other types of concrete. This is due to the fact that fibers were not included in these compositions. UHPC mixtures typically contain high quality steel fibers, allowing for partial (and in some cases full) replacement of the transverse steel reinforcement while also helping with control cracking. Considering that savings in steel reinforcement are not accounted for with the current indices developed herein, steel fibers were included in all mixtures evaluated and their impact is show in figure 9. This allows comparing the efficiency indicators for the proposed members in applications where the use of high quality steel fibers is required/beneficial (e.g. improving flexural strength and ductility in pavement overlay [6], or increasing the load at which cover spalling occurs in heavy-loaded columns [101]). Figure 9 illustrates the significant impact associated with using steel fibers in all the different types of concrete compared to figure 7. This figure suggests that, for the calculated indices and for applications that require the use of high-quality steel fibers, higher strength concretes (especially UHPC) are generally more efficient. This is even more evident in the plots associated with cost-efficiency indicators. This can be explained with the fact that high quality steel fibers are relatively expensive. For a typical content of 1.5% (by volume), steel fibers can add $650 m −3 to a given mix composition [85]. This increases the unit cost of material in UHPC by over a factor of 2.3, while increasing this cost in NSC by a factor of approximately 8. Additionally, steel fiber production entails high volumetric carbon footprints (see table 1). The combination of high specific gravity of steel (7.8 g m −3 ) with the high volumetric carbon footprint of these fibers (0.88 kg.CO 2 -eq m −3 ) [90] result in significant contributions to the environmental impact of the cast member, in spite of the low volume of fibers typically used (<2%). Consequently, and as shown in figure 9 for the initial cracking application, the use of fibers significantly impacted the eco-efficiency index for the concrete technologies requiring higher volumes of material per structural element. These results should encourage and urge further research on optimizing fiber content in UHPC, where small changes could have considerable cost and sustainability implications. Additionally, further research should be conducted to find alternative high-ductility, high-tensile-strength fibers to reduce the cost and environmental impact of UHPC products.
The main message from these scatter plots is that, with proper optimization, advanced concrete materials such as HPCs and UHPCs can be the most eco-friendly and cost-efficient solutions for new infrastructure and superstructures, in applications where dimensional reduction of cross-sections meets constructability limitations. For instance, it may be difficult to justify these materials in small infrastructure, where the reductions in dimensions of the structural elements can only go as low as constructability permits (e.g. cross sectional area of small columns and beams as a function of required steel). On the other hand, superstructural elements such as bridge girders can greatly benefit from the outstanding mechanical properties of UHPC, especially considering proportional increases in prestressing levels based on increases in compressive and tensile strengths. Such an increase in prestressing levels can result in longer spans, reduced cross-sectional areas (lighter sections) and fewer girder lines. This not only reduces the number of superstructural elements, but recursively reduces the number of substructural elements (columns, piers and foundations elements), which require large volumes of concrete. UHPC also reduces the amount of steel required (as a result of reduced section sizes and the potential to more effectively utilize higher strength reinforcement), which is extremely important considering the embodied emissions associated with this material. Such advantages are not yet captured in the analysis presented herein. New indices that account for the impact of steel reinforced concrete on cost and eco-efficiency must be further developed to elucidate these arguments for applications involving UHPC. The path has been set with the initial efforts made by Kourehpaz and Miller [19], where indices that account for steel in design have been developed. Yet, these indices were developed for simple cases while keeping the required area of steel constant, which inhibits accurate comparisons between NSC and UHPC solutions.

Conclusions
The concept of sustainable design involves environmental, economic and health, and community impacts. Knowing that decisions made during structural design and construction phases directly affect all these facets, mixture design selection should target a balanced optimization function to accommodate all three of these objectives. As governments and policy makers push towards low carbon policies, it is important to develop tools to promote holistic design processes that meet performance and sustainability requirements in a cost-efficient way. In addition, there is a lack of proper environmental indicators to demonstrate the effect of mechanical performance and mix proportions, in dictating the dimensions of structural elements and associated costs and embodied carbon dioxide (CO 2 ) emissions.
Accordingly, this study proposed a methodology that enables the effect of mix proportioning on performance, cost and CO 2 emissions to be evaluated concurrently utilizing multi-objective ML-based tools and multi-criteria comparison indices. Key findings from this work are: • MODDs developed in this study have shown to be ideal tools for global concrete mixture optimizations.
These tools enable cost and eco-efficiency considerations during mix design selection while satisfying performance thresholds required by design. MODDs provide designers and concrete producers with alternative solutions to properly consider material availability and the use of locally sourced materials; • The weighted allocations assigned to cost and eco-efficiency greatly impact the determination of optimum solutions. For multi-objective functions, iterations may be performed to find the most balanced solutions for both objectives. In this sense, MODDs can serve as proof of optimization to help justify the tradeoffs considered between cost and emissions when selecting a given mixture for a particular application; • A case study was presented to compare the contributions of concrete to the environmental impact of short columns considering mixtures optimized for volumetric indicators against mixtures optimized for efficiency metrics. Results show that the mixture with the lowest volumetric CO 2 did not produce the column with the lowest total embodied CO 2 , which was obtained when optimizing for the eco-efficiency index. Similar observations were made for cost; • The case study also demonstrates the impact associated with the weights allocated to cost and eco-efficiency.
For a typical 15-story reinforced concrete building with 300 columns, the most cost-efficient mix design (MO-Col-1) was compared to the most eco-efficient mix design (MO-Col-6). Results show that while selecting MO-Col-1 allows savings of approximately $12 700 compared to MO-Col-6, it also produces an excess of 100 tons CO 2 -eq compared to MO-Col-6. These results demonstrate how optimization studies focused solely on cost can have tremendous environmental consequences with marginal financial gains; • The proposed multi-criteria indices were also used to compare different types of concretes from the literature as a function of binder content and compressive strength. Results show that there is no clear correlation between binder content and eco-efficiency indices. The same result was observed with respect to compressive strengths. This indicates that high paste content, high strength (and ultra-high strength) concrete technologies are not necessarily detrimental to cost or eco efficiencies. In fact, for the cases evaluated, optimum solutions were mostly obtained with these types of concrete; • Results showed a weak correlation between volumetric impact indicators and efficiency indices for the short column application. On the other hand, there is a stronger correlation between volumetric impact indicators and efficiency indices for initial cracking applications. This is mostly due to the reduced influence of compressive strength on cracking indices. Yet, there is still a visible shift/offset in the observed trends for different types of material, which is associated with increased compressive strengths and increased differences in the modulus of rupture between these different concretes. Therefore, cost and eco-efficiency comparisons between different types of concrete should not be performed on a volumetric impact basis; • The effects of incorporating 1.5% by volume of steel fibers (typical content for UHPC) was evaluated for all the mixtures considered in this study. Results suggest that, for the indices evaluated and applications requiring use of high-quality steel fibers, higher-strength concretes are generally more efficient in terms of cost and carbon footprint. This is explained by the fact that these fibers are relatively expensive and entail high embodied CO 2 . Thus, concrete technologies that allow reducing the dimensions of structural elements directly reduce the volume of fibers used.
The trends observed in this study suggest that, for the construction industry, thresholds on carbon footprint should preferentially be imposed on total GHG emissions associated with a given structural member type (bending, compressive, etc) and project instead of a per volume basis. The totality of results indicate that poor mix design solutions can be obtained with any concrete technology. Yet, with proper optimization strategies, advanced concrete materials such as UHPC can be the most cost-effective and environmentally sustainable solution for new infrastructure in applications where reduction in dimensions meets constructability constraints. Additionally, MODDs have the potential to be game-changing mix design tools. The makeup of MODDs facilitates the evaluation of rather complicated trends associated with mix proportions and multi-objective outcomes, allowing ML-based tools to be of easy interpretation by industry personnel with no expertise in AI. MODDs could be used as aids in the decision-making process during mix design stages and serve as proof of mixture optimization that could be introduced in EPDs.
Future work should focus on developing new indices that account for the environmental impact of reinforcing steel on eco-efficiency and further development of ML-based design tools for material properties other than compressive strength.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https://doi. org/10.18738/T8/5MPO52. Figure A1. PDD describing the predicted compressive strengths for varying contents of concrete sand, crushed sand and ground quartz (Reproduced from [21]. CC BY 4.0).