A goal programming approach for balancing life-cycle carbon burden and lean value in Industry 4.0 portfolio selection

Value concept within the lens of lean manufacturing

Within lean manufacturing, value is defined primarily from the customer perspective and is created by meeting customer requirements while minimizing non-value-adding activities and unnecessary resource consumption. Lean thinking therefore emphasizes systematically reducing waste, variability, and overburden across production processes.

Industry 4.0 technologies can complement these principles by improving process visibility, responsiveness, data availability, and operational integration^[28-30]. Real-time monitoring, analytics, automation, and digitally enabled coordination can support reductions in variability, downtime, defects, lead times, and inventory while improving resource utilization and flexibility.

Accordingly, lean value is treated in this study as a multidimensional construct rather than a single financial measure^[31]. Five performance dimensions are considered: (i) lead-time reduction; (ii) defect and rework reduction; (iii) work-in-process and inventory reduction; (iv) resource efficiency improvement; and (v) flexibility and responsiveness. These dimensions provide the operational basis for the lean value representation used in the proposed model.

Carbon footprint implications of Industry 4.0 technologies

Industry 4.0 technologies can generate carbon burdens across several stages of their life cycle, including hardware production, operation, maintenance, and end-of-life treatment^[9,10,32]. A life-cycle perspective is therefore important because restricting assessment to operational energy use alone may overlook material- and equipment-related impacts occurring before and after deployment^[33].

Operational carbon is particularly relevant for data-intensive technologies. AI, cloud computing, digital twins, and related applications may require substantial computing, storage, communication, and supporting infrastructure^[20]. Recent carbon-aware optimization studies illustrate this issue in digital infrastructure. Hanafy et al. examine carbon–cost–performance trade-offs in cloud workload scheduling^[34], Cao et al. incorporate computing and physical-infrastructure emissions into adaptive data-center capacity provisioning^[35], and Zhao et al. jointly optimize energy cost and carbon footprint in cloud data centers^[36]. These studies show that the carbon burden associated with digital technologies depends not only on operation but also on the supporting infrastructure required for deployment and use.

Production-stage carbon is also relevant because sensors, edge devices, robotics, mobile platforms, and semiconductor-based equipment require materials, component manufacturing, assembly, and associated supply-chain activities^[37,38]. Maintenance, replacement, and end-of-life treatment can add further burdens over the technology’s service period, particularly for electronic components and batteries.

At the same time, Industry 4.0 technologies may enable indirect carbon reductions through improved energy efficiency, waste prevention, predictive maintenance, and process optimization. Such avoided emissions, however, depend on plant-specific baselines, operating conditions, and interactions among technologies and cannot be inferred directly from the technology itself. The present study therefore distinguishes between potential technology-enabled reductions and the technology-attributable carbon burden associated with adopting and operating the technologies. The latter defines the carbon-side decision boundary used in the portfolio model, while technology-enabled operational improvements are represented through the lean value dimension.

Trade-off structure, GP, and research positioning

Taken together, the preceding discussion indicates that Industry 4.0 investment decisions involve a portfolio-level trade-off between operational value creation and the carbon burden associated with technology adoption. Technologies that improve process visibility, quality, flexibility, resource utilization, and responsiveness may simultaneously require additional hardware, computing, communication, and energy resources^[6,7,10]. Technology selection should therefore account for both operational benefits and environmental burden rather than treating either dimension in isolation.

GP provides an appropriate structure for this type of decision when managers can specify aspiration levels for competing objectives. Instead of optimizing a single criterion, GP minimizes deviations from predefined targets and can therefore represent situations in which carbon control and operational improvement are pursued simultaneously. Previous applications have used GP in sustainable and low-carbon supply-chain planning, project and portfolio decisions, and lean production system implementation^{[22,23,26,39-41]}. These studies establish GP as a well-established multi-objective decision method; accordingly, the contribution of the present study does not arise from introducing a new GP technique.

The contribution of the present study instead concerns the integration of several features within Industry 4.0 portfolio planning. Candidate technologies differ in their technology-attributable carbon burden, lean value contribution, investment requirement, implementation difficulty, and dependence on enabling infrastructure^[20,21,38]. Existing Industry 4.0 studies provide important approaches for technology assessment and selection, while related optimization studies address environmental technology portfolios or lean-oriented objectives. To clarify the position of the present framework relative to these streams, Table 1 compares six closely related studies using the decision features directly relevant to the proposed model.

The comparison shows that these studies address complementary parts of the present decision problem. Almeida et al. provide a multidimensional framework for Industry 4.0 investment appraisal and explicitly recognize technology prerequisites, but do not formulate them as portfolio constraints^[12]. Maretto et al. extend technology selection to groups of related digital technologies^[15], whereas de Mendonça Santos et al. prioritize sustainability functions supported by Industry 4.0 technologies rather than optimizing technology portfolios^[16]. Kellner et al. optimize powertrain technology portfolios under greenhouse gas (GHG) considerations^[42], while Bonello et al. combine life-cycle environmental assessment with multi-objective optimization of technology combinations^[11]. Karakutuk and Ornek, in turn, demonstrate the use of GP for lean-oriented production objectives^[23].

Among the related studies examined in Table 1, none simultaneously represents portfolio selection, technology-attributable carbon burden, lean-oriented value, implementation capacity, and technology prerequisites within the same prescriptive decision model. The contribution of the present study is therefore integrative and problem-oriented rather than a methodological extension of GP. The framework brings these decision dimensions together in a binary weighted GP formulation that identifies feasible Industry 4.0 portfolios under carbon and lean aspiration levels, resource limitations, and technology-dependency conditions.

Problem definition

The decision problem addressed in this study concerns the selection of an Industry 4.0 technology portfolio that simultaneously supports lean-oriented value creation and controls the technology-attributable carbon burden associated with emerging digital and automation technologies. A firm is assumed to evaluate a finite set of candidate technologies, each of which requires a given investment cost, consumes a certain implementation capacity, contributes to lean value creation, and generates a technology-attributable, life-cycle-oriented carbon burden. The central managerial question is therefore not only which technologies provide higher operational value, but also which combination of technologies can be adopted without creating an excessive carbon burden.

The problem has an inherently multi-objective structure. On the one hand, the firm seeks to minimize the total carbon burden of the selected technologies. On the other hand, it aims to maximize lean-oriented value creation through improvements in lead time, quality, inventory, resource efficiency, and operational flexibility. In its basic form, the decision structure can be expressed as follows:

subject to budgetary, implementation-capacity, technology-dependency, and binary selection constraints. These two objectives may conflict with each other. A portfolio that generates higher lean value may require additional hardware, more energy-intensive digital infrastructure, or greater computational capacity, whereas a lower-carbon portfolio may fail to achieve the desired level of lean transformation. Because the two criteria are measured in different units and reflect different managerial priorities, optimizing either objective in isolation would provide an incomplete basis for strategic technology selection.

To reconcile these competing priorities, the proposed framework adopts a GP approach. In the proposed model, carbon burden and lean value are treated as target-based goal dimensions: exceeding the carbon target and falling short of the lean value target are the undesirable deviations to be minimized. The model therefore seeks a technology portfolio that minimizes the weighted and normalized deviations from these two targets, subject to budget, implementation-capacity, and dependency constraints.

Model assumptions

To delineate the scope of the proposed framework and to keep the formulation tractable and transparent, the following assumptions are adopted:

(1) Each candidate technology is either fully adopted or not adopted within the planning horizon; partial or incremental adoption is not considered, which justifies the use of binary decision variables.

(2) All model parameters, including carbon components, lean value contributions, investment costs, and implementation-capacity requirements, are treated as deterministic point estimates in the baseline formulation. Uncertainty in these inputs is examined separately through the robustness analysis.

(3) In the baseline formulation, portfolio-level carbon footprint and lean value are calculated from technology-level parameters. These parameters are interpreted as adjusted, incremental, and non-overlapping decision inputs rather than as gross, standalone effects. Accordingly, the lean value contribution of a technology represents its additional contribution after accounting for the roles of prerequisite and enabling technologies, while the carbon parameter reflects the technology-attributable burden assigned to that technology, without double-counting shared infrastructure. Interaction effects beyond the explicitly modeled prerequisite and infrastructure dependencies are not estimated endogenously in the baseline model.

(4) The decision is made for a single strategic planning period; the timing and sequencing of investments within the horizon are not modeled.

(5) The aspiration levels for carbon burden and lean value, as well as the goal weights, are specified by the decision-maker and reflect the firm’s strategic orientation toward sustainability and operational excellence.

These assumptions define the baseline formulation and keep the model transparent for target-based portfolio selection. Extensions for portfolio-level interaction effects are discussed after the model formulation.

Construction of carbon parameter

The carbon parameter C_i represents the technology-attributable carbon burden of technology i. The total carbon burden is represented through four components:

where C_i^prod denotes production-stage carbon associated with raw material acquisition, component manufacturing, hardware production, assembly, packaging, transport, and initial deployment; C_i^op represents operational carbon arising from electricity use, data processing, cloud computing, communication, and computational requirements; C_i^maint captures the carbon burden associated with servicing, spare parts, replacement activities, and hardware or software maintenance; and C_i^eol represents end-of-life carbon associated with recycling, recovery, treatment, or disposal.

The system boundary covers the carbon burden attributable to the technologies themselves. The model does not estimate the net plant-level carbon effect after accounting for technology-enabled energy savings, waste reduction, defect prevention, reduced downtime, or other avoided emissions. Such operational improvements are represented, where relevant, through the lean value dimension and are not subtracted from C_i. This separation avoids treating lean resource-efficiency improvements as measured carbon reductions or implicitly crediting avoided emissions without plant-specific evidence. The framework therefore identifies which portfolio delivers lean value under a specified technology-attributable carbon burden; it does not determine which portfolio minimizes the net carbon footprint of the plant.

Construction of lean value parameter

Lean-oriented value creation is defined as a composite lean value score derived from multiple lean performance dimensions. Let q_im denote the illustrative lean contribution score of technology i with respect to lean value dimension m. In the numerical application, q_im is assigned on a 1-10 scale, where higher values indicate stronger expected contribution. Let α_m be the weight of lean value dimension m, with α_m > 0 and Σ_m∈Mα_m = 1. The composite lean value score of technology i is calculated as:

Since q_im ∈ [1, 10], the weighted average Σ_m∈Mα_mq_im also lies in [1, 10], and L_i lies in [10, 100]. The multiplication by 10 is used only to express the technology-level lean contribution on a convenient 10-100 score scale.

In a firm-specific application, both the dimension weights and technology scores should be elicited systematically rather than assigned directly. A cross-functional expert panel representing relevant functions such as production, maintenance, quality, information technology, and sustainability can first define anchored descriptions for the 1-10 scoring scale. The dimension weights α_m may then be elicited using a structured weighting method such as analytic hierarchy process (AHP) or best–worst method (BWM), while the technology-level scores q_im may be obtained through a Delphi-based procedure, in which experts provide independent initial assessments, review anonymized group feedback, and revise their judgments where appropriate. The resulting scores may be aggregated using the median or another prespecified group-aggregation rule.

As with any expert-based assessment, the elicited values may be affected by anchoring, differences in scale interpretation, expert selection bias, or overly optimistic expectations for particular technologies. These risks can be reduced through explicit scale anchors, independent initial scoring, anonymized feedback, and a common evaluation procedure applied consistently across all technologies. The resulting parameters should therefore be interpreted as structured managerial judgments, and their uncertainty can be examined through sensitivity or robustness analysis.

Sets, parameters, and decision variables

The notation used in the proposed model is summarized below.

Sets
I     Set of candidate Industry 4.0 technologies, indexed by i.
M     Set of lean value metrics, indexed by m.
P     Set of prerequisite relationships; if (j, i) ∈ P, technology j may be selected only if prerequisite technology i is also selected.
A_j     Set of alternative infrastructure technologies, any one of which can satisfy the infrastructure requirement of technology j.

Parameters
C_i     Total technology-attributable carbon burden of technology i.
C_i^prod     Production-stage carbon components of technology i.
C_i^op     Operational carbon components of technology i.
C_i^maint     Maintenance-related carbon components of technology i.
C_i^eol     End-of-life carbon components of technology i.
L_i     Composite lean value score of technology i.
q_im     Lean contribution score of technology i with respect to metric m.
α_m     Weight of lean metric m.
K_i     Investment cost of technology i.
H_i     Implementation-capacity requirement of technology i.
B     Total available investment budget.
H^max     Maximum available implementation capacity.
C^target     Carbon target.
L^target     Lean value target.
w_C, w_L     Weights of the carbon and lean value goals.

Decision variables
x_i     Binary variable equal to 1 if technology i is selected and 0 otherwise.
d_C⁺, d_C^-     Positive and negative deviations from the carbon target.
d_L⁺, d_L^-     Positive and negative deviations from the lean value target.

Binary weighted GP formulation

Following the standard weighted GP structure described by Jones and Tamiz^[45], the proposed binary formulation is expressed as follows:

subject to:

The objective function (5) minimizes the weighted and normalized deviations from the carbon footprint and lean value targets. Normalization by the corresponding target makes the two deviation terms dimensionless and prevents differences in numerical scale from determining their relative importance. Only d_C⁺ is penalized for the carbon goal because exceeding the carbon aspiration level is undesirable, whereas only d_L^- is penalized for the lean goal because falling short of the desired lean value is undesirable. Performance beyond either aspiration level receives no additional reward. This target-satisfaction structure represents a satisficing decision context in which the objective is to control carbon burden while attaining a desired level of lean value^[22,24]. If continued improvement beyond either target is itself valued, a different objective structure, such as a conventional weighted objective or a lexicographic formulation, would be more appropriate.

The weights w_C and w_L represent the relative importance assigned to proportional carbon excess and proportional lean value shortfall. They should therefore not be interpreted as a direct exchange rate between one CFU and one unit of lean value. Because portfolio decisions are discrete, changes in the weights do not necessarily produce gradual changes in the solution; Section “Weight breakpoints and Pareto-efficiency validation” examines the resulting weight intervals and portfolio breakpoints.

Constraints (6) and (7) are the goal constraints; they measure, respectively, the deviation of the selected portfolio’s total carbon footprint and total lean value from their target levels. Constraint (8) ensures that total investment cost does not exceed the available budget, while constraint (9) limits the cumulative implementation difficulty of the selected portfolio to the firm’s available managerial and technical capacity. Constraint (10) enforces strict prerequisites, while constraint (11) requires that, for any technology depending on supporting infrastructure, at least one admissible infrastructure alternative be present. Finally, conditions (12) and (13) impose the binary and non-negativity requirements on the decision and deviation variables, respectively.

The baseline formulation assumes additive portfolio-level carbon and lean value. To avoid double counting, the technology-level parameters are therefore constructed as incremental, non-overlapping contributions relative to a defined reference configuration of the plant’s existing sensing, network, computing, and analytics infrastructure. For dependent technologies, the contribution is assessed conditional on the prerequisite technology already being available. For example, the lean contribution of predictive maintenance is evaluated as the additional improvement obtained when IoT-based monitoring is already in place. Shared infrastructure is accounted for only once, either as a separate platform component or through allocation among the technologies using a documented physical basis such as computing time, storage demand, communication traffic, or installed sensing capacity. The same incremental principle is applied to shared carbon burdens.

Where material interactions remain after this attribution procedure, the formulation can be extended using pairwise interaction variables y_ij, where y_ij = 1 if technologies i and j are selected together, and y_ij = 0 otherwise. Let S denote the set of technology pairs for which interaction effects are relevant. The auxiliary variables are linearized as:

The corresponding portfolio-level lean value and carbon burden become:

where γ_ij represents synergy, overlap, or diminishing returns in lean value, and δ_ij represents additional or avoided carbon burden associated with joint implementation or shared infrastructure. These interaction coefficients are not estimated in the numerical application because they require firm-specific deployment and operational data. The baseline model therefore uses the incremental technology-level parameters together with the explicit dependency structure.

Scenario-based preference setting

To examine how managerial priorities shape the selected Industry 4.0 technology portfolio, the model is solved under three preference scenarios. These scenarios represent different strategic orientations toward carbon control and lean value creation, with w_C + w_L = 1:
Carbon-priority: w_C = 0.70, w_L = 0.30
Balanced: w_C = 0.50, w_L = 0.50
Lean-priority: w_C = 0.30, w_L = 0.70

The carbon-priority scenario gives greater emphasis to avoiding carbon-target violations. The balanced scenario assigns equal importance to carbon control and lean value creation. The lean-priority scenario emphasizes achieving the desired lean value level while still penalizing carbon excess. Comparing the resulting portfolios provides a scenario-based assessment of how the selected technology set changes as the strategic emphasis shifts between sustainability and operational improvement.

Technology alternatives

The firm evaluates ten Industry 4.0 technology alternatives, listed in Table 2. These technologies were selected to represent a broad set of digital, data-driven, and automation-oriented technologies commonly associated with smart manufacturing and lean transformation. Prior studies on the Industry 4.0 and lean manufacturing interface emphasize that technologies such as IoT, cloud computing, simulation, advanced robotics, additive manufacturing, and data analytics can support different levels of the value stream and contribute to operational improvement, process visibility, and resource efficiency^[28-30]. Recent studies also highlight a broader technology set, including AI, industrial IoT, digital twins, big data analytics, advanced robotics, and additive manufacturing, as key enablers of sustainable and data-driven industrial transformation^[10].

The alternatives differ in their technological role within the illustrative setting. IoT-based monitoring (T1), radio frequency identification (RFID)/real-time location system (RTLS) tracking (T9), and edge computing (T8) are treated primarily as enabling infrastructure technologies because they support data acquisition, connectivity, traceability, and local processing. Predictive maintenance (T3), digital twin (T6), and cloud manufacturing analytics (T7) are represented as data-driven decision-support technologies because they use operational data to improve planning, monitoring, maintenance, and process control. Collaborative robots (T4), automated guided vehicle (AGV)/autonomous mobile robot (AMR) internal logistics (T5), and additive manufacturing (T10) are treated as more hardware-intensive technologies.

Parameter setting for the numerical application

The numerical application uses a structured illustrative parameter set covering carbon burden, lean value contribution, investment cost, implementation difficulty, managerial targets, and technology dependencies. Each parameter is classified according to its construction basis as literature-informed (L), engineering-based (E), or application-specific (X). Literature-informed parameters reflect relative technology characteristics reported in previous studies; engineering-based parameters represent comparative technical characteristics applied consistently across the alternatives; and application-specific parameters represent values that would be estimated or elicited from firm data, managers, or experts in an empirical implementation. Table 3 summarizes this classification. Because the construction basis represents comparative technology profiles rather than empirical absolute levels, the parameters should be interpreted as structured relative decision inputs. Their numerical magnitudes are used directly in the optimization, while the absolute values should not be interpreted as measured firm-level quantities.

Illustrative carbon parameters

The carbon parameters are constructed comparatively across the ten alternatives using the four components defined in Section “Construction of carbon parameter” and the construction bases summarized in Table 3. The component values for each technology are reported in Supplementary Table 1.

Figure 1 presents the resulting carbon composition of the alternatives. Hardware-intensive technologies exhibit comparatively larger production-stage components, whereas data- and computing-intensive technologies exhibit relatively greater operational components. The alternatives therefore differ in both their carbon profiles and their total technology-attributable carbon burdens.

Figure 1. Carbon footprint components of technology alternatives. CFU: Carbon footprint unit; IoT: Internet of Things; AI: artificial intelligence; AGV: automated guided vehicle; AMR: autonomous mobile robot; RFID: radio frequency identification; RTLS: real-time location system.

Illustrative lean value parameters

Lean-oriented value creation is represented through the five performance dimensions defined in Section “Construction of lean value parameter”. Technology-level scores are assigned on a common 1-10 scale to represent relative contributions to lead-time reduction, quality improvement, work-in-process and inventory reduction, resource efficiency, and flexibility and responsiveness. Equal dimension weights (α_m = 0.20) are adopted as the baseline weighting structure, and the composite lean value follows from Equation (4). The scores are structured to represent heterogeneous lean contribution profiles across the five dimensions. Supplementary Table 2 reports individual scores and the resulting composite values.

The resulting lean values are composite decision parameters rather than measured performance outcomes. In an empirical implementation, firm experts would elicit both the technology scores and the dimension weights using the procedure described in Section “Construction of lean value parameter”. Figure 2 summarizes the weighted and rescaled lean value components of the alternatives, consistent with the composite lean value calculation in Equation (4).

Figure 2. Weighted and rescaled lean value components of the technology alternatives. IoT: Internet of Things; AI: artificial intelligence; AGV: automated guided vehicle; AMR: autonomous mobile robot; RFID: radio frequency identification; RTLS: real-time location system; WIP: work-in-process.

Technology dependency constraints

Industry 4.0 technologies may rely on shared data-acquisition, computing, localization, or tracking infrastructure. These relationships are represented in the model through the strict prerequisite and infrastructure-alternative constraints introduced in Section “Proposed GP model formulation” [Equations (10) and (11)]. The numerical application adopts the architecture-specific dependencies reported in Table 5.

These constraints represent the technology architecture assumed for the illustrative application rather than universal requirements of the corresponding technologies. In a firm-specific application, the dependency set should be defined according to the organization’s existing digital infrastructure and implementation architecture, and individual constraints may therefore be added, removed, or modified. The robustness analysis in Section “Robustness Analysis” evaluates the sensitivity of the results to alternative relaxed and strengthened dependency configurations.

Computational results

The model was implemented in General Algebraic Modeling System (GAMS) and solved separately for each preference scenario. Because the application contains only ten candidate technologies, global optimality was independently verified by complete enumeration of all candidate portfolios. Table 6 reports the optimal portfolios and their performance values.

The optimal portfolio changes systematically with managerial preference. In the carbon-priority scenario (S1), the model selects {T1, T3, T6, T7}. This portfolio has a total carbon burden of 215 CFU, which is below the carbon target of 220 CFU, while its portfolio lean value is 296, leaving a lean value shortfall of 54. Thus, the portfolio avoids carbon excess but does not fully attain the lean value target.

In the balanced scenario (S2), the optimal portfolio is {T1, T3, T6, T8, T9}, yielding a carbon burden of 233 CFU and a portfolio lean value of 334. Relative to S1, this portfolio accepts a carbon excess of 13 CFU while reducing the lean value shortfall from 54 to 16. It therefore represents an intermediate solution between carbon control and lean value creation. In the lean-priority scenario (S3), the optimal portfolio is {T1, T3, T6, T7, T9}. This portfolio meets the lean value target exactly at 350 but increases the carbon burden to 251 CFU, resulting in a carbon excess of 31 CFU. Under the specified budget and implementation-capacity limits, attaining the lean value target is therefore associated with a higher carbon burden. Figure 3 compares the carbon burden and portfolio lean value of each solution with their respective targets.

Figure 3. Portfolio performance against the carbon and lean value targets across the three scenarios. (A) Carbon burden vs. target; (B) Portfolio lean value vs. target. CFU: Carbon footprint unit.

Across the three scenarios, the trade-off pattern is clear. As the preference emphasis shifts from carbon control toward lean value creation, carbon excess increases from 0 to 13 and then to 31 CFU, while the lean value shortfall decreases from 54 to 16 and then to 0, as shown in Figure 4. Resource utilization also increases in the same direction. Budget use rises from 285 to 320 and then to 330 CU, corresponding to 84%, 94%, and 97% of the available budget. Implementation-capacity use increases from 16 to 17 and then to 18 units, with the lean-priority portfolio fully utilizing the available implementation capacity.

Figure 4. Scenario outcomes in terms of (A) carbon excess and (B) lean value shortfall. CFU: Carbon footprint unit.

As a validation check, all possible portfolios were enumerated. After filtering by budget, implementation capacity, and dependency constraints, 125 feasible portfolios remained, 31 of which were Pareto-efficient. The three GP solutions were all on this efficient frontier, confirming that no dominated portfolio was selected. The enumeration was used only for verification, while the GP model remains the main decision-support framework.

Across the three nominal scenarios, T1, T3, and T6 are selected repeatedly, whereas T7, T8, and T9 vary with the preference structure. These patterns reflect the illustrative parameter set and dependency assumptions. Their stability under simultaneous parameter uncertainty is examined in the robustness analysis presented in Section “Robustness Analysis”.

Sensitivity analysis

A sensitivity analysis was conducted to examine whether the proposed model produces interpretable and consistent portfolio responses under changing preference, target, and resource conditions. Because the numerical application is based on an illustrative parameter set, the sensitivity analysis does not validate empirical parameter estimates; it evaluates the model’s behavior under alternative assumptions.

Four sensitivity analyses are reported. SA1 varies the preference weights; SA2 varies the carbon target; SA3 varies the investment budget; SA4 varies the budget and the carbon target jointly. Table 7 summarizes SA1-SA4.

Preference weight sensitivity

In the first experiment, the preference weights assigned to the carbon and lean value goals were varied while keeping the carbon target, lean value target, budget, and implementation capacity fixed. The results reveal three distinct portfolio regions. When the model assigns greater weight to the carbon goal, it selects the carbon-conservative portfolio {T1, T3, T6, T7}. This portfolio keeps the total carbon burden below the target but leaves a lean value shortfall of 54. As the preference structure becomes more balanced, the model shifts to the compromise portfolio {T1, T3, T6, T8, T9}, which accepts a carbon excess of 13 CFU while reducing the lean value shortfall to 16. When greater weight is assigned to the lean value goal, the model selects {T1, T3, T6, T7, T9}, attaining the lean value target while increasing the carbon excess to 31 CFU.

As illustrated in Figure 5, the model responds consistently to changes in managerial priorities. A stronger emphasis on carbon control leads to lower-carbon portfolios, whereas a stronger emphasis on lean value creation leads to portfolios with higher lean value but also higher carbon burden. This result supports the central premise of the numerical application: under the assumed parameter setting, Industry 4.0 portfolio selection involves an explicit trade-off between carbon control and lean-oriented value creation. Supplementary Table 3 reports the optimal portfolio, goal deviations, and resource use at each weight setting.

Figure 5. Sensitivity to managerial preference weights. (A) Carbon excess; (B) Lean value shortfall. CFU: Carbon footprint unit.

Carbon target sensitivity

In the second experiment, the carbon target was varied under the balanced preference setting. The results indicate that stricter carbon targets make lower-carbon technology combinations more attractive, but this may come at the expense of lean value performance. For example, when the carbon target is reduced to 180 CFU, the selected portfolio produces a lean shortfall of 74. As the carbon target is relaxed, the model can include additional technologies that improve lean value performance. When the carbon target reaches 260 CFU, both the carbon and lean value goals are achieved without any undesirable deviation.

Figure 6 shows that the carbon target functions as a strategic control parameter rather than a passive reporting threshold. A stricter target limits the attractiveness of technology combinations with higher carbon burdens, even when they offer higher lean value contributions. Conversely, a more flexible carbon target allows the firm to adopt a broader technology portfolio and reduce the lean value gap. Therefore, the carbon target can be interpreted as a managerial lever for aligning digital transformation intensity with environmental ambition. The full results for all carbon target levels and preference scenarios are reported in Supplementary Table 4.

Figure 6. Sensitivity to the carbon target (balanced setting). (A) Carbon excess; (B) Lean value shortfall. CFU: Carbon footprint unit.

Budget sensitivity

In the third experiment, the available investment budget was varied while maintaining the balanced preference setting. The results show that increasing the budget initially broadens the feasible solution space and improves lean value performance. When the available budget increases from 220 to 320 CU, the lean shortfall decreases from 126 to 16. This indicates that the budget constraint is influential when financial resources are limited. However, after the budget reaches 320 CU, further increases do not alter the selected portfolio. At budget levels of 320 CU and above, the selected portfolio uses 320 CU, leaving positive budget slack for higher budget levels.

As shown in Figure 7, additional financial resources improve the selected portfolio only up to the point at which the budget constraint ceases to be binding. Beyond this point, a firm cannot necessarily obtain a better Industry 4.0 portfolio by simply increasing its investment budget. Under the balanced weighting scheme and the baseline carbon target, reducing the remaining lean value shortfall would require selecting a more carbon-intensive portfolio, thereby increasing carbon excess and worsening the weighted goal-deviation objective. Thus, financial resources are effective only when they align with the carbon target and implementation capacity. The complete budget sweep is reported in Supplementary Table 5. Supplementary Table 6 summarizes the saturation diagnosis for each preference scenario, including the resources used at the saturation point and the objective penalty of the best remaining single addition.

Figure 7. Sensitivity to the investment budget (balanced setting). (A) Carbon excess; (B) Lean value shortfall. CFU: Carbon footprint unit; CU: cost unit.

Robustness analysis

The sensitivity analyses in Section “Sensitivity analysis” vary managerial settings while holding the technology-level parameters fixed. In an empirical application, these parameters would also be subject to estimation uncertainty. A Monte Carlo analysis was therefore conducted to examine the stability of the selected technologies and portfolios under simultaneous parameter perturbation.

The experiment combines two perturbation levels (ρ = 0.10 and ρ = 0.20), three dependency configurations, and the three preference scenarios defined in Section “Scenario-based preference setting”, yielding 18 experimental cells. Each cell contains 10,000 replications. Within each preference scenario, the goal weights remain at their nominal values, while the budget and target levels are held at the baseline settings.

In each replication, the four carbon components, five lean sub-scores, investment costs, implementation-difficulty values, and implementation-capacity limit are multiplied by independent factors drawn from Uniform(1 - ρ, 1 + ρ). Lean sub-scores are restricted to the admissible range [1, 10], while implementation-difficulty and capacity values are retained as continuous normalized quantities. Lean dimension weights are drawn from a Dirichlet distribution centered on the baseline equal-weight vector. The concentration parameter is set to ν = 399 for ρ = 0.10 and ν = 99 for ρ = 0.20, corresponding to an approximate coefficient of variation of 10% and 20% for each dimension weight.

Technology dependencies are examined under three alternative configurations rather than being perturbed randomly. The baseline configuration uses the dependencies in Table 5. The relaxed configuration assumes that a condition-monitoring layer is already available and therefore removes the three prerequisite relationships involving T1. The strengthened configuration adds a computing requirement for the digital twin and restricts the computing alternative for AI-based quality inspection to edge computing. Detailed definitions of the three configurations are provided in Supplementary Table 8.

Each replication was evaluated by exhaustively enumerating all candidate portfolios, ensuring that the reported variation reflects parameter uncertainty rather than solver tolerance. The complete design therefore comprises 180,000 solved instances. Table 9 reports technology selection frequencies averaged across the three dependency configurations for each perturbation-level–preference-scenario combination, together with the minimum and maximum frequencies observed across all 18 experimental cells. Figure 8 complements these results by showing the nominal selection pattern and the minimum, maximum, and mean selection frequencies across the 18 robustness cells. The complete matrix is provided in Supplementary Table 9.

Figure 8. Nominal technology selection and selection-frequency ranges under parameter uncertainty. (A) Selection in the nominal scenarios; (B) Range across the 18 robustness cells. IoT: Internet of Things; RFID: radio frequency identification; RTLS: real-time location system; AI: artificial intelligence; AGV: automated guided vehicle; AMR: autonomous mobile robot.

IoT-based machine monitoring (T1) and predictive maintenance (T3) exhibit the highest selection stability. Across the 18 cells, T1 is selected in 87.2% to 100% of replications and T3 in 80.7% to 99.5%. The high frequency of T1 is partly structural, since it serves as a prerequisite for three other technologies under the baseline and strengthened configurations; its frequency decreases when these relationships are removed in the relaxed configuration. T3, by contrast, is not a prerequisite for another technology and remains highly selected across the alternative dependency structures.

The three dependency configurations yield broadly similar exact-reproduction rates. Holding the perturbation level and preference scenario fixed, exact reproduction of the nominal portfolio varies by at most 4.5 percentage points across the configurations. The clearest technology-level effect concerns T1, whose selection frequency decreases under the relaxed configuration when its prerequisite role is removed. This confirms that part of T1’s apparent stability is structural rather than attributable solely to its carbon and lean value parameters. Apart from this effect, the alternative dependency configurations produce relatively small changes in the overall portfolio-stability pattern.

The digital twin (T6), cloud manufacturing analytics (T7), edge computing (T8), and RFID/RTLS tracking (T9) show greater sensitivity to the perturbed parameters and preference scenarios. Across these four technologies, selection frequencies span 29.9% to 98.7% across the experimental cells. In particular, T6, although selected in all three nominal portfolios, varies between 58.2% and 82.0%, indicating greater parameter sensitivity than T1 and T3. AI-based quality inspection (T2), collaborative robots (T4), AGV/AMR internal logistics (T5), and additive manufacturing (T10) are selected in no more than 11.0% of replications in any cell. These results apply to the perturbation ranges and dependency configurations examined and should not be interpreted as general rankings of the technologies.

The robustness results therefore qualify the nominal interpretation in Section “Computational results”. T1 and T3 exhibit the highest cross-scenario selection stability, whereas the repeated selection of T6 in the nominal solutions is more sensitive to the assumed parameter values. The identification of candidate core technologies is consequently dependent not only on the nominal optimization results but also on their stability under parameter uncertainty.

Portfolio composition is less stable than technology-level selection. At ρ = 0.10, the nominal portfolio is reproduced in approximately 61% to 67% of replications for the carbon-priority and balanced scenarios and about 33% for the lean-priority scenario. At ρ = 0.20, exact reproduction rates fall to approximately 16% to 35% [Supplementary Table 10]. Thus, individual technology selection patterns remain more stable than the exact composition of the complete portfolio when multiple inputs are perturbed simultaneously.

Weight breakpoints and Pareto-efficiency validation

The effect of preference weights and the efficiency of the goal-programming solutions can be examined exactly because all candidate portfolios can be enumerated. For a fixed portfolio, the objective is linear in the lean preference weight w_L, with w_C = 1 - w_L. The optimal portfolio therefore changes only at intersections between portfolio-specific objective functions. Over the open interval (0 < w_L < 1), three optimal portfolio regions are obtained, as reported in Table 10 and illustrated in Figure 9.

Figure 9. Weight regions and breakpoints of the optimal portfolio.

At w_L = 0.3524, the first and second portfolios are co-optimal, while at w_L = 0.6415, the second and third portfolios are co-optimal. The endpoints represent limiting cases: at w_L = 0, only carbon excess is penalized, and 83 feasible portfolios attain zero deviation, whereas at w_L = 1, the lean-priority portfolio is the unique optimum. The balanced setting (w_L = 0.50) lies farther from a breakpoint than the carbon-priority (w_L = 0.30) and lean-priority (w_L = 0.70) settings, with distances to the nearest breakpoint of approximately 0.142, 0.052, and 0.058, respectively.

Complete enumeration yields 31 Pareto-efficient portfolios in the carbon–lean objective space. All three goal-programming solutions belong to this efficient set, confirming that none is dominated under the baseline parameter setting. Supplementary Table 11 reports the complete frontier.

Only 6 of the 31 efficient portfolios are supported efficient solutions; the remaining 25 are unsupported and therefore cannot be generated by a conventional positive weighted-sum scalarization of portfolio carbon and lean value. The carbon-priority solution T1 + T3 + T6 + T7, with 215 CFU and a lean value of 296, is one of these unsupported solutions. The goal-programming formulation can reach such a portfolio because it minimizes one-sided deviations from aspiration levels rather than a linear weighted sum of the two portfolio objectives.

GP does not generally guarantee Pareto efficiency because improvements beyond an aspiration level may remain unpenalized. The solutions obtained here are efficient by direct enumeration. Where a goal-programming optimum is dominated, or multiple solutions share the same primary objective value, a secondary Pareto-refining optimization can be applied after fixing the optimal goal-programming value. An ε-constraint formulation could likewise generate the efficient frontier; complete enumeration is used here because the 10-technology illustrative application contains only 2¹⁰ =1,024 candidate portfolios, making exact evaluation computationally tractable.

How managers can implement this model in practice

The framework can be implemented through a five-step decision process. First, managers define the decision scope by specifying the plant or value stream, planning horizon, candidate technologies, and the existing digital infrastructure that forms the reference configuration. Second, technology-level inputs are assembled. Carbon parameters should be estimated on a consistent system boundary and planning-horizon basis using available firm-specific, supplier, or life-cycle data, while lean contribution scores and dimension weights can be elicited using the procedure described in Section “Construction of lean value parameter”. Investment costs and implementation-capacity requirements should likewise reflect the firm’s own deployment conditions.

Third, managers specify the decision constraints and aspiration levels. This includes the available investment budget and implementation capacity, the technology dependencies associated with the firm’s existing architecture, and the carbon and lean value targets. Carbon coefficients and the carbon target should be expressed on the same accounting basis.

Fourth, the model is solved under alternative managerial preference settings rather than relying on a single weight combination. Scenario solutions and the weight-breakpoint analysis in Section “Weight breakpoints and Pareto-efficiency validation” can then be used to identify the portfolio changes associated with different strategic priorities.

Finally, the robustness analysis described in Section “Robustness Analysis” can be applied using uncertainty ranges appropriate to the quality of the available data. Technologies that remain frequently selected across plausible parameter and dependency settings can then be distinguished from selections that are more sensitive to the underlying assumptions. The model inputs can subsequently be updated as firm-specific implementation and operational data become available.

Table 11 summarizes the principal findings and their managerial implications.