Multi-Aspect Probability Model of Expected Profit Subject to Uncertainty for Managerial Decision-Making in Local Transport Problems
Abstract
1. Introduction
- What types of costs and revenues should governments consider to make economically efficient decisions in the short and long term for the benefit of the whole society?
- In the presence of uncertainty, how should governments evaluate road remediation projects to maximize benefits for society?
- Short-term model (SM)
- Long-term-short-term model (LSM)
- Social long-term-short-term model (SLSM)
- Long-term-short-term model extended by a time aspect (TLSM)
- Development of Comprehensive Models: We propose four probability-based models that provide a robust framework for evaluating road remediation projects under uncertainty. These models incorporate a wide range of costs and benefits, including short-term and long-term factors, social costs, and time-related costs.
- Integration of Uncertainty: The models explicitly address the issue of uncertainty by using probabilistic principles to express expected costs and profits. This allows decision-makers to better understand the potential risks and rewards associated with different project options.
- Partial ex post Validation: The concepts are partially validated through an ex post analysis of a past road project in Slovakia. This provides empirical evidence of the models’ utility for multi-criteria decision-making in transportation problems.
- Focus on Societal Benefit: The study emphasizes the importance of considering societal costs and benefits in addition to purely economic factors. This ensures that decisions are made in the best interests of the entire community.
2. Theoretical Research Implementation
3. Materials and Methods
- 1. Short-term model (SM): This model focuses on the immediate costs and revenues of road remediation projects, providing a short-term perspective on economic efficiency.
- 2. Long-term-short-term model (LSM): Expanding on the SM model, the LSM model incorporates long-term considerations, allowing for a more comprehensive evaluation of costs and benefits over an extended period.
- 3. Extensions of LSM model: This section introduces two extensions of the LSM model, further refining the economic evaluation process.
- 3.1. Social extension: SLSM model: The SLSM model incorporates social costs into the analysis, providing a broader societal perspective on the impacts of road remediation projects.
- 3.2. Time extension: TLSM model: Recognizing the value of time, the TLSM model incorporates time-related costs and benefits into the evaluation, providing a more nuanced understanding of the economic efficiency of road remediation projects.
- RQ1: What types of costs and revenues should the government take into account to make an economically efficient decision in a short time period and in a long time period for the whole society?
- RQ2: Given uncertainty, how should the government evaluate a transport problem of road remediation to maximize the profit for society?
- The cost–benefit analysis [29]—This concept of economic analysis can be applied in various areas when an economic actor decides between multiple options subject to uncertainty; the result is supposed to be economically optimal for the actor.
- Application of theory of expected profits adapted from [30]—Since it is not possible to explicitly determine some phenomena in the future, our model uses probabilistic form, and the expected mean value of the given variable is always taken into account.
3.1. Short-Term Model (SM)
- SR is a random variable which expresses the amount of extra euros that the government collects due to the increased amount of refueling caused by closures, detours, or restrictions. Moreover, SR is the sum of extra value-added tax and extra consumption tax collected by the government as a result of road closures and related detours.
- b is a value of the random variable SR.
- is the probability that the government will collect taxes in the amount of b EUR.
- is a probability density function of the random variable SR.
- RC is a random variable which expresses direct costs of reconstruction (repair costs).
- FC is a random variable which expresses direct costs of financial compensation.
- DC is a random variable which expresses damage costs.
- r is a value of the random variable RC.
- c is a value of the random variable FC.
- d is a value of the random variable DC.
- is the probability that the remediation will cost r EUR.
- is the probability that the compensations will cost c EUR.
- is the probability that the damage sanitation will cost d EUR.
- are probability density functions of the random variable RC, FC, DC.
3.2. Long-Term-Short-Term Model (LSM)
- SR is a random variable which expresses the amount of extra euros that the government collects due to the increased amount of refueling caused by closures, detours, or restrictions. Moreover, SR is the sum of extra value-added tax and extra consumption tax collected by the government as a result of road closures and related detours.
- b is a value of the random variable SR.
- is the probability that the government will collect taxes in the amount of b EUR.
- is a probability density function of the random variable SR.
- I+ is a random variable which expresses an increase in investment in the region compared to the current situation as a result of the implementation of the given variant (in EUR).
- L+ is a random variable which expresses an increase in saved lives (number of people) compared to the current situation as a result of the implementation of the given variant.
- i is a value of the random variable I+.
- l is a value of the random variable L+.
- is the probability that the government will additionally receive an investment in the amount of i EUR in the future.
- is the probability that the government will save l people the future as a result of the implementation of variant vi.
- is a probability density function of the random variable I+.
- is a probability density function of the random variable L+.
- w is the average amount of taxes that one person pays to the government during their lifetime.
- SR, I+, L+, RC, DC, FC are random variables.
- b, i, l, r, d, c are values of the random variables SR, I+, L+, RC, DC, FC which we defined above.
- is the probability that a random variable VAR = {SR, I+, L+, RC, DC, FC} is equal to the value of k = {r, c, d, f}.
- are probability density functions of the random variables SR, I+, L+, RC, DC, FC.
- SR, I+, L+, RC, DC, FC are random variables.
- bj, ij, lj, rj, dj, cj are values of the random variables SR, I+, L+, RC, DC, FC.
- is the probability that a discrete random variable VAR = {SR, I+, L+, RC, DC, FC} is equal to the value of xj.
- are probability functions of the discrete random variables SR, I+, L+, RC, DC, FC.
3.3. Extensions of LSM Model
3.3.1. Social Extension: SLSM Model
- F is a random variable which expresses the amount of extra euros that the drivers have to pay for extra fuel due to the increased amount of refueling caused by closures, detours, or restrictions.
- f is a value of the random variable I+.
- is the probability that the drivers will pay f EUR because of purchases of extra fuel.
- is a probability density function of the random variable F.
- RC, DC, FC, F are random variables.
- r, f, d, c are values of random variables RC, FC, DC, F which we defined above.
- fk are denoted as the probability density functions of these random variables, k = {r, c, d, f} and
- is the probability that a random variable VAR = {SR, I+, L+, RC, DC, FC} is equal to the value of k = {r, c, d, f}.
- are probability density functions of the random variables RC, FC, DC, F.
3.3.2. Time Extension: TLSM Model
- TS is a random variable which expresses the amount of time that one driver/passenger saves thanks to realization of the selected variant vi.
- s is a value of the random variable TS.
- is the probability that a driver saves s amount of time.
- is a probability density function of the random variable TS.
- z is a value of the random variable TL, which is defined as the amount of time that a driver loses due to the realization of the selected variant vi.
- TL is a random variable which expresses the amount of time that a driver loses due to the realization of the selected variant vi.
- z is a value of the random variable TL.
- is the probability that a driver loses z amount of time.
- is a probability density function of the random variable TL.
- TS, TL are random variables.
- s, z are values of random variables TS, TL which we defined above.
- is the probability that a random variable VAR = {TL, TS} is equal to the value of k = {s, z}.
- are probability density functions of the random variables TS, TL.
- TS is a random variable which expresses the amount of time that one driver/passenger saves thanks to realization of the selected variant vi.
- s is a value of the random variable TS.
- is the probability that a driver saves s amount of time.
- is a probability density function of the random variable TS.
- SR, I+, L+, RC, FC, DC, F, TS, TL are random variables.
- r, f, d, c are values of random variables RC, FC, DC, F which we defined above.
- fk are denoted as the probability density functions of these random variables, k = {r, c, d, f}.
- is the probability that a random variable VAR = {SR, I+, L+, RC, FC, DC, F, TS, TL} is equal to the value of k = {b, i, l, r, c, d, f, s, z}.
- are probability density functions of the random variables SR, I+, L+, RC, FC, DC, F, TS, TL.
- n1 is a constant related to the sum of the number of drivers/passengers passing this route per given period, n2 is related to the number of drivers/passengers that have to use the detour caused the reconstruction in a given time period, and m is the average wage per hour.
4. Results
- First, we found out how many days the route under Strecno was closed.
- We then estimated the number of vehicles which had been diverted from the original route due to remediation—we found out how many days the route under Strecno was closed.
- We calculated the number of extra kilometers.
- Finally, the total extra fuel consumption in EUR was quantified as the sum of extra fuel consumption by personal vehicles, trucks, and motorcycles.
- According to results from every route, we estimated the extra costs to society (costs which drivers had to incur due to the closure of route 1/18).
- 6.
- Partial restriction, two-way road in two choked lines (width of 3 m), the speed limit set to 30 km per hour.
- 7.
- Total closure, i.e., the transition through the passage was restricted and was guarded by police.
- 8.
- 11 weekdays (from Monday to Friday) and 36 weekends
- 9.
- During weekends, the time of closure was set from Saturday 8 am until Sunday 4 pm.
- 10.
- During weekdays, it was a 24 h closure except for one day when the passage was closed only 3 h
- 11.
- In sum, the road was closed for 823 h.
5. Discussion
- The long-term-short-term model (LSM) expands the concept of the SM model by taking into account long-term benefits, such as the increase in investment in the region and the increase in saved lives, compared to the current situation as a result of the implementation of the given variant.
- The social long-term-short-term model (SLSM) extends the LSM model by taking into account the costs to society (drivers) that arose due to increased fuel consumption caused by the implementation of detours.
- The long-term-short-term model with time aspect (TLSM) takes into account the time lost due to the implementation of the reconstruction and the further time gained thanks to the implementation of the given solution.
Limitations and Future Research
- Data Availability: The study relies on estimations and average figures for certain parameters, such as the number of vehicles affected by road closures, which may not reflect the actual situation accurately. This reliance on estimations can affect the precision of the results and their ability to provide accurate implications in real-world scenarios.
- Regional Specificity: The models are validated using a specific road project in Slovakia, and their direct applicability to other regions with different geographical, economic, and social contexts may be limited. Factors such as terrain, traffic patterns, and economic activity can vary significantly between regions, potentially influencing the outcomes of road remediation projects.
- Project Scope: The study focuses primarily on road remediation projects, and the models may not be directly applicable to other types of transport projects, such as new road construction or public transportation initiatives. Each type of project has unique characteristics and considerations that may require adjustments to the models.
- Uncertainty Simplification: While the models incorporate uncertainty using probabilistic functions, they may not capture all types of uncertainty that can influence real-world outcomes. Factors such as political decisions, unforeseen events, or changes in economic conditions can introduce uncertainties that are not fully accounted for in the models.
- A limitation may also be the fact that, in practice, the probability density of a given random variable is unknown. This poses a problem in terms of solving the proposed models, since, in order to quantify individual variants, we need to have defined probability densities of individual random variables (e.g., the probability of collecting an additional amount of tax for all possible values of tax collection in EUR). One option to solve this is to approximately estimate these functions based on expert estimates in cooperation with experts. Another option is to use expert knowledge to create several possible scenarios of the development of the specified variant and calculate the expected values of random variables as discrete random variables, as stated in relation (18).
- Moreover, a model with a time aspect is also a limitation, since in this model, in order to be able to quantify it in monetary terms, it is necessary to know constants such as the number of people who will pass the given section during a period of time, the average wage, the reconstruction period, and also the number of years that we will take into account for calculating profits. In practice, this may be difficult to determine, and incorrect parameter settings may distort the results. This model may also be complicated by the fact that the number of people transported is not always equal to the number of cars, since there may be more than one person in a car or bus. This would also need to be considered in the time aspect model. In the case of drivers, the situation could be easier to estimate; however, it is the number of passengers which is more appropriate as the input number in the model.
- Other limitations also concern the validation of our concepts on the defined problem of road 1/18 in Slovakia. In this validation, we based our study on defined concepts rather than models in the true sense, since we did not have defined individual relevant probability (density) functions of random variables. However, since this was a validation of a solution that had already been implemented, we decided to partially investigate it using our concepts. However, during the validation itself, we did not have all the data available, and therefore, we did not include the variables I+ and L+ in the LSM model. Since this was only a partial road reconstruction, we assume that the I+ variable would not be significantly affected. However, in terms of lives saved, it is possible to assume that the reconstruction would have a certain impact on this variable, which was not quantified. Moreover, in the validation calculations, we used a large number of constants, some of which were only approximate and some that could be preliminary (e.g., the share of trucks, the average number of cars per day, the tax rate, or not taking into account the differences in the ratio of cars/trucks on weekdays and weekends). Setting the constants to other, more accurate values could also change the results compared to those we presented. However, rather than accurately quantifying the presented situation under Strecno Castle, we wanted to show the thought process of quantifying individual solutions to traffic problems. In the validation part, we also did not take into account unofficial detour routes, which could also change the results.
- Adaptation to Other Sectors: Future research could explore adapting the proposed models to other sectors beyond transportation, such as healthcare or environmental management, where economic evaluation under uncertainty is crucial.
- Integration of More Complex Uncertainty: Future studies could investigate incorporating more complex types of uncertainty into the models, such as using fuzzy logic or scenario planning to capture a wider range of potential outcomes.
- Refinement of Social Cost Estimation: Further research could refine the estimation of social costs by considering additional factors, such as environmental impacts or the distribution of costs and benefits across different population groups.
- In the future, it would be possible to focus on further expanding the models to include other aspects, or on thorough quantification of the model with a time aspect.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Category | Type | Description | Model(s) |
---|---|---|---|
Costs | Reparation Costs | Costs directly related to road repair or remediation. | SM, LSM, SLSM, TLSM |
Financial Compensation Costs | Costs associated with compensating individuals or businesses affected by the road project (e.g., due to accidents or detours). | SM, LSM, SLSM, TLSM | |
Damage Sanitation Costs | Costs incurred due to accidents or damage to property related to the road condition. | SM, LSM, SLSM, TLSM | |
Social Costs | Additional costs incurred by society due to the project, such as increased fuel consumption due to detours. | SLSM | |
Time Costs | Value of time lost by drivers/passengers due to road closures, detours, or delays. | TLSM | |
Benefits | Extra Revenues | Additional tax revenues collected due to increased (extra) fuel consumption resulting from the project. | SM, LSM, SLSM, TLSM |
Increased Investment (in EUR) | Long-term benefits from increased investment in the region due to improved infrastructure. | LSM, SLSM, TLSM | |
Lives Saved | Tax value of lives saved due to improved road safety (related to tax payments in EUR) | LSM, SLSM, TLSM | |
Time Saved | Value of time saved by drivers/passengers due to improved road conditions and reduced travel times. | TLSM |
Time | Dist. Loss | Length | Time Loss | ||
---|---|---|---|---|---|
route A | section Strečno | 31 min | 0 | 29.3 km | 0 |
detour route B | detour across Zázriva | 75 min | −44 km | 78.8 km | −49.5 min |
detour route C | detour across Trenčín | 162 min | −131 km | 197 | −167.7 min |
constants | number of vehicles per day (in km) | n | 26,831.00 |
number of hours closed | h | 823.00 | |
number of days closed | u | 34.29 | |
truck rate | g | 0.33 | |
official detour for trucks (in km) | dt | 131.00 | |
official detour for cars (in km) | dc | 44.00 | |
average fuel price | fp | 1.50 | |
taxation rate for fuels | t | 0.50 | |
calculations | sum of extra km for cars (in km) | kmv1 | 790,977.88 |
sum of extra km for trucks (in km) | kmv2 | 1,159,904.13 | |
sum of extra km total (in km) | kmt | 1,950,882.01 | |
SM model | extra revenue into state budget (in EUR) | 1,463,161.51 | |
reconstruction costs (in EUR) | 1,100,000.00 | ||
financial compensation (in EUR) | N/A | ||
damage sanitation (in EUR) | N/A | ||
sum of short-term costs (in EUR) | SC | 1,100,000.00 | |
SM model short-term profit (in EUR) | SP | 363,161.51 | |
LSM model | extra revenue into state budget (in EUR) | SR | 1,463,161.51 |
future investments | N/A | ||
lives saved | N/A | ||
reconstruction costs (in EUR) | 1,100,000.00 | ||
financial compensation (in EUR) | N/A | ||
damage sanitation (in EUR) | N/A | ||
sum of long-term short-term costs (in EUR) | SC | 1,100,000.00 | |
LSM model long-term profit (in EUR) | TPLSM | 363,161.51 | |
SLSM model | extra revenue into state budget (in EUR) | 1,463,161.51 | |
future investments | N/A | ||
lives saved | N/A | ||
reconstruction costs (in EUR) | 1,100,000.00 | ||
financial compensation (in EUR) | N/A | ||
damage sanitation (in EUR) | N/A | ||
Costs to society | 2,926,323.02 | ||
SLSM total costs | TCSLSM | 4,026,323.02 | |
SLSM model profit (costs to society included) (in EUR) | TPSLSM | −2,563,161.51 |
Model | Time Horizon | Scope | Key Factors | Best Use Cases |
---|---|---|---|---|
Short-term Model (SM) | Short-term | Immediate costs and benefits | - Reparation costs - Financial compensation costs - Damage costs—Extra revenues | Projects with limited timeframe and immediate impact, such as urgent road repairs or temporary traffic management solutions. |
Long-term-Short-term Model (LSM) | Long-term | Extended costs and benefits over time | - All factors in SM - Increased investment - Lives saved | Projects with long-term implications, such as major road construction or infrastructure development. |
Social Long-term-Short-term Model (SLSM) | Long-term | Societal costs and benefits | - All factors in LSM - Social costs (e.g., increased fuel consumption due to detours) | Projects with significant social impacts, such as road closures or changes in traffic patterns affecting communities. |
Long-term-Short-term Model with Time Aspect (TLSM) | Long-term | Time-related costs and benefits | - All factors in LSM - Time saved - Time lost | Projects where time is a critical factor, such as improving traffic flow or reducing travel times. |
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Holubčík, M.; Falát, L.; Soviar, J.; Dubovec, J. Multi-Aspect Probability Model of Expected Profit Subject to Uncertainty for Managerial Decision-Making in Local Transport Problems. Logistics 2025, 9, 39. https://doi.org/10.3390/logistics9010039
Holubčík M, Falát L, Soviar J, Dubovec J. Multi-Aspect Probability Model of Expected Profit Subject to Uncertainty for Managerial Decision-Making in Local Transport Problems. Logistics. 2025; 9(1):39. https://doi.org/10.3390/logistics9010039
Chicago/Turabian StyleHolubčík, Martin, Lukáš Falát, Jakub Soviar, and Juraj Dubovec. 2025. "Multi-Aspect Probability Model of Expected Profit Subject to Uncertainty for Managerial Decision-Making in Local Transport Problems" Logistics 9, no. 1: 39. https://doi.org/10.3390/logistics9010039
APA StyleHolubčík, M., Falát, L., Soviar, J., & Dubovec, J. (2025). Multi-Aspect Probability Model of Expected Profit Subject to Uncertainty for Managerial Decision-Making in Local Transport Problems. Logistics, 9(1), 39. https://doi.org/10.3390/logistics9010039