Characterisation of basin water allocation benefit function using a sigmoid-type S-curve logistic equation

According to WHO (2013), a human being requires a minimum of 15–20 L of water per day for basic short-term survival. Meanwhile, for medium-term lifestyle maintenance, up to 70 L of water will be required daily. The requirements follow after the Sphere Humanitarian Charter and Minimum Standards in Disaster Response, which set standards for the minimum level of services people affected by an emergency should receive. Hence, the non-discretionary water use can be defined as the water that is required to meet the basic consumptive and sanitation needs. Meanwhile, discretionary uses are non-essential or optional (Willis et al. 2011). However, there is an increasingly large proportion of discretionary components in essential water-use activities, which constitute wastage (Russell & Knoeri 2020). The ‘basic benefit’ of water supply is thus fully realised at a much lower threshold, whereas a higher supply level serves to meet the convenience, lifestyle and elasticity of discretionary demands.

According to Gilbertson et al. (2011), significantly more people from water-scarce locations are supportive of most water conservation behaviours. The finding suggests that the experience of drought influences an individual's attitude to water conservation. Nonetheless, the ‘stickiness’ of water-saving behaviours very much depends on behavioural, psychosocial and contextual factors (Dean et al. 2021). The primary reasons are the serious implications for efficient operations and the different perceptions of individual well-being, or ‘benefit’ (Callejas Moncaleano et al. 2021).

Studies on the decision support system (DSS) for water allocation (WA) often aim to balance the stakeholders' requirements and the local context concerning the economic, social and environmental impact of water use (e.g., Ma & Zheng 2011; Sjöstrand et al. 2020; Nel et al. 2022). Popular approaches are to maximise the combined economic benefit for all water users and/or the community to ensure the economic sustainability of the allocation system, while considering the social drivers and ecological and environmental priorities, among others. However, quantifying the ‘benefit’ is a non-trivial task (e.g., Selelo et al. 2017). Quantitative estimates of the consequences of financial losses due to short- and long-term water disruptions are still scarce (Sjöstrand et al. 2021). Consequently, most studies resort to a simple scoring system ranked by expert judgement or targeted respondents. It is thus desirable to have an established and broadly accepted mathematical description of water-use benefit, which can be readily applied for analysis in a WA DSS.

The phenomenon of S-curve is ubiquitous throughout nature and the man-made world. It can be seen in plant studies (e.g., Gregorczyk 1991; Kawano et al. 2020), grain yield (Zhang et al. 2019), technology changes (Fisher & Pry 1971), innovation in education (Hipkins & Cowie 2016), construction (Tijanić Štrok & Car-Pušic 2017; Konior & Szóstak 2020), project management (Mohamad et al. 2021), carbon dynamic (Yu & Yea 2020), economic growth (Smirnov & Wang 2017), water shortage risk (Qian et al. 2018) and many other fields, such as population changes, market penetration, social inventions, and ecological modelling. Despite its limitation in long-term prediction, the forecasting power of an S-curve is attributed to the basic concept of limited resources that lie at the basis of any growth process, lending it a popular model for describing the evolution of systems (Kucharavy & De Guio 2011).

In a seminal work which was headlined ‘seeing the S-curve in everything’ by Phys.org (2011), Bejan & Lorente (2010) explain the reason for the prevalence of this pattern using the constructal law. According to them, all phenomena (i.e., of ‘everything’), the animate or the inanimate, are configured to flow and move as a conglomerate of engine and brake designs. The ‘engine’ part evolves towards generating more power, whereas the ‘brake’ part evolves towards more dissipation. The result is an S-curve behaviour, which is characterised by a slow initiation period, an accelerated middle phase and a drawn-out consolidation final stage.

An early attempt at the qualitative use of the S-curve pattern to describe the benefit of water use was reported by Zuo (2008) and Florke et al. (2013). Sjöstrand et al. (2021) investigated time-dependent water supply resiliency factors for economic sectors by focusing on the level of output that businesses can sustain during water disruption. Their results show an S-curve trend with an elongated tail, but the curves vary between sectors due to the different levels of dependency on the water supply.

In another global-scale study on potential water scarcity impacts by Dolan et al. (2021), a log-modulus function is presented to quantify the economic impact as the difference between the constrained and unlimited economic surplus. They considered data over the 21st century for 235 river basins for each of the 3,000 global change scenarios, which are simulated using the Global Change Analysis Model-integrated assessment model. Physical water scarcity is defined as the withdrawal-to-availability ratio. The basins, which are relatively water-rich, are the ones with the highest number of positive impact scenarios.

Besides the impact on the economy, water shortages also have a detrimental effect on public health and human development, especially the vulnerable groups, such as the elderly, children and women (Tarrass & Benjelloun 2012). Water insecurity negatively affects the ability of households to adhere to protective public health measures compared to households that have access to continuous supply (Kumpel et al. 2022). According to Majuru et al. (2016), the choice of coping strategies, due to unreliable water supplies, is influenced by, among others, income and level of education. The findings showed that low-income households bear a disproportionate coping burden, which are characterised by labour- and time-intensive strategies, which yield smaller quantities and the lower quality of water, thus exposing them to greater health risks.

The literature above showed that the S-curve phenomenon is already observed in the social and economic aspects of WA. However, there are no established benefit function curves that can be used by water managers. For operational WA, the ability to quantify these benefits in numeric values that can be readily incorporated into the analysis process for objective decision-making is sought. Ideally, the mathematical description of the benefit level or impact, where applicable, should be directly correlated to the WA as a fraction of the maximum benefit under the full supply level. The generic framework would thus be portable and highly adaptable to cater for varied objectives and purposes. In this paper, a questionnaire is designed to measure the perceived social impact, hence benefit, of reduced WA to water users. The economic benefit, meanwhile, is assessed based on the correlation between the GDP and the actual water consumption. Data obtained are subjected to mathematical fitting using a sigmoid-type, or logistic-type, equation. A generalised logistic benefit function curve is then proposed as a standard tool to quantify WA benefits in water-related management decision-making.

A questionnaire was conducted to evaluate the impact, hence benefit, of water reduction to water users. Google form is used and the questionnaire is circulated through WhatsApp social media to different user groups from diverse backgrounds, primarily in the Selangor state in Malaysia.

The introduction of the questionnaire is designed to educate the respondents before they answer the questions. First, it is mentioned that the prevalent water supply design in Malaysia adopts a minimum design value of 225 L per day (LPD). Next, it is highlighted, according to WHO (2013), that individuals need only 15–20 L for basic survival (drinking and cooking), which is less than 10% of the supply level. It is further explained that one-third of the supply level (<70 L) is adequate for general household hygiene, sanitation and cleaning purposes. The aim is to ensure that the respondents are fully aware of the non-discretionary and discretionary components of water use, notwithstanding the generous supply level enjoyed.

Respondents are then requested to rate their perceived impact if the water supply is reduced under five different allocation WA scenarios, as shown in Table 1, with the respective description.

Respondents are then required to select from a linear scale of 0 to 10 representing the benefit levels, where 0 indicates affected, whereas 10 is not affected. There are only five responses required, corresponding to the five supply scenarios. Hence, the questionnaire is simple and straightforward and can be completed within less than a minute.

Following Guo et al. (2022), it is assumed that economic development can be correlated to water use. The GDP data of Malaysia by state are acquired from the Department of Statistics, Malaysia for the 8-year period between 2015 and 2022. The GDP value is based on the 2015 constant price in million Ringgit Malaysia (RM).

The total water consumption of the states for the same time is obtained from the Annual Factbook published by the National Water Services Commission (SPAN 2022), the regulatory body for the water and sewerage industry for Peninsular Malaysia and the Federal Territory of Labuan. This refers to the reticulated treated water supply only and does not include other water resources such as groundwater which constitute a larger fraction of the water consumption in states such as Kelantan. The data presented for the state of Selangor are inclusive of the Federal Territories of Kuala Lumpur (KL), which is located inside the state.

The scatter plot of state GDP, as a function of their respective total water consumption, herein referred to as the economic benefit of water consumption (EBWC), is then produced for further analysis.

The logistic function, or more generally, the sigmoid function, is a broad class of functions which exhibit an S-shaped curve, mapping input values to a range between 0 and 1. It was first introduced as a model of population growth (Kucharavy & De Guio 2011). While a bell-shaped curve describes the rate of growth of a life cycle of a system with limited resources, the cumulative growth follows an S-curve, which illustrates the different phases of birth, growth, maturity, decline and death for any system.

According to Kucharavy & De Guio (2015), inappropriate use of S-shaped curves can lead to strange and inadequate results. They proposed and illustrated the use of a single logistic curve and logistic component analysis focusing on the coherence between model, data and interpretation.

As the constants vary, the benefit is not zero or one at the lower and the upper bounds of the water availability (e.g., curves iv and v).

The questionnaire survey was carried out for 1 month. The total number of responses to the questionnaire is 130. Table 2 shows the summary of the mean benefit B_i for each WA scenario, with the signature of an S-curve. The result suggests that respondents perceived an immediate ‘impact’ of reduced WA in Scenario 1 even though the actual reduction is small and has a negligible tangible effect on day-to-day discretionary water use. For the reduction to 10% of the design supply value (Scenario 5), most respondents maintain that they are not yet ‘terribly affected’ (average score 0.06 > 0), which is likely based on the clarification that the amount is suffice for short-term survival needs.

According to Fan et al. (2014), water users’ perceived water consumption and actual water consumption are influenced by factors, such as gender, income, education level and water conservation consciousness. There is a general tendency to underestimate outdoor and kitchen water consumption and overestimate indoor water consumption. This suggests that the perceived impact of water reduction may be biased by a sense of volume-to-space ratio. In the present survey results, a reduction (volume) from a prevalent condition (space) is more felt than a ‘further’ reduction under the scarcity condition (reduced space), which explains the diminishing impact felt.

The uncertainties and the associated confidence levels in S-curve logistic fits are functions of the uncertainty on the data points and the length of the data period (Debecker & Modis 1994). They recommended minimisation of the chi-squared distribution to better determine the three parameters a, b and c, which define the curve.

Kucharavy & De Guio (2011) reported many instances of S-curve applications that omit the vertical scale (qualitative) or adopt arbitrary units. They highlighted the importance of carefully selecting appropriate measurable parameters to derive meaningful interpretations from the approach. In our present case, the vertical limits are defined using B = 1 for maximum benefit and B = 0 for maximum impact, but it is not possible to impose the same on the logistic function. Besides, according to Debecker & Modis (1994), points near the extremities (<5 and >95%) should not be expected to fit well, which is similarly observed in our present result.

Debecker and Modis also suggested that the data used for S-curve fitting should be a minimum of 20% of the full range of the S-curve. Here, our data points spread 80% of the range but only at five distinct values along the horizontal axis. The steepness, which characterised the mid-section of the S-curve, is thus approximated from limited data points. A similar attempt, to develop an S-curve based on a small number of points in a time series, has been reported by anticipating the logistic development of the phenomenon, hence its inflection point (midpoint) and the saturation level (Rządkowski & Sobczak 2020).

As shown in Figure 5, the benefit function fitted the survey response much better, with R² = 0.998. The derived S-curve trajectory is in excellent agreement with the data points and maintains a reasonable approximation of the upper and lower bounds.

From the fitted curve, the disturbed zone, the perturbed zone, the distressed zone and the distraught zone as WA reduces can be identified. The turning point is located approximately at WA = 0.7 and the diminishing point at WA = 0.3, respectively.

Using Equation (2), the best logistic regression is then determined but keeping a = 1 and b = 10. For the two cases A and B considered, the R² values obtained are 0.8796 (c = −4.7) and 0.6181 (c = −5.2), respectively (Table 4), which are a slight improvement over the equation proposed by Lee et al. (2023). Furthermore, the upper and lower bounds are sought. The fitted c values for the two cases are quite consistent: the upper bound is given by c = −3.8 and −3.4, and the lower bound is given by c = −7.5 and −7.2, respectively (Figure 8).

Compared to the early growth phase, the accelerated growth phase shows a higher increase in EBWC, which suggests a higher translational effect from water use to economic gain. This may be a result of higher water-use efficiency in the various economic sectors, e.g. irrigation, industries, etc.

The curve proposed by Lee et al. (2023) grossly overestimated the rate of economic return in the accelerated growth phase. The curve also underestimated the further growth potential in the consolidated stage. As a region prospers, further economic growth is more likely to be driven by a water-independent economy in lieu (e.g., services) and thus may see GDP rise which decouples from total water consumption.

Finally, based on the assumption that the GDP is directly related to the day-to-day business operation and revenue generation, the logistic regression derived may thus be extended to represent the impact of reduced WA to the economic sectors during water scarcity. The longer the period affected by the water cut, the more likely the economic impact approaches that described by the trend.

A sigmoid-type logistic equation is derived in this study to fit the perceived benefit and impact of reduced WA to water users. The result of the questionnaire survey on the social benefit of WA comprises four distinct supply-and-benefit states, namely, ‘disturbed’, ‘perturbed’, ‘distressed’ and ‘distraught’, in the order of increasing severity due to water shortage. The benefit function approaches unity at the full supply condition but approaches nought if the water supply is completely cut off.

The EBWC based on 8-year data of states in Malaysia shows evidence of the S-curve characteristics, which can be characterised by the early economic growth phase, middle accelerated growth phase and advanced consolidated growth phase. The less developed states tend to benefit more as water consumption increases. The effect exhibits increased benefit returns at higher consumption rates and eventually reflects the switch towards further growth driven by a water-independent economy. It is herein posited that the S-curve derived can be used to describe the impact of WA reduction on the economy sector.

In summary, this paper shows that the logistic-type function is a good approximation of the social and economic benefits of water supply and can be used to provide quantitative assessment in water-related decision-making.

The authors acknowledge the invaluable feedback from the anonymous reviewers of the early manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.