Multiobjective Intuitionistic Fuzzy Optimization Approach in Optimal Irrigation Planning and Operation of Reservoir

A Multiobjective Intuitionistic Fuzzy Linear Programming model is developed to get an optimal cropping pattern in the command area of the Ukai-Kakrapar Irrigation project in Gujarat, India. Four conflicting objectives, namely, maximization of net irrigation benefits (NIB), maximization of employment (EG), minimization of cultivation cost (CC), and maximization of revenue generation on account of industrial and municipal supplies (MI) were optimized for their crisp linear programming (LP) solutions. Intuitionistic Fuzzy Optimization Multi-objective fuzzy linear programming (IFO MOFLP), IFO MOFLP with hesitation index, and two-phase IFO MOFLP models were developed from their LP solutions. The TPIFO MOFLP model has been found to perform better than other variant of IFO MOFLP models and the MOFLP model (Average operator Case-I) developed by Mirajkar and Patel in J Water Resour Plan Manag 142(11):1–16, 2016 for the same command area with reference to the uncertainty parameters like degree of acceptance (α), rejection (β), and hesitation index (π) for 75% dependable inflow condition. The recommended TPIFO MOFLP model also gives higher irrigation intensity (112.19%) than Average Operator Case-I (104.60%) for the whole command area. The implementation of TPIFO MOFLP model in the command area is likely to give the values of NIB, EG, CC, and MI as Rs 10,836.19 million, 34,980.4 thousand workdays, Rs 5,672.23 million, and Rs 2,314.03 million, respectively, with α = 0.68, β = 0.19, and π = 0.13. The suggested model with new parameters like α, β, and, π, would assist the decision makers to apply the same on real world problems with greater certainty.


Introduction
Water, a precious life-source, is getting scarce due to global climatic and anthropogenic changes. Accelerated growth of population and industries in recent years has increased its demand manifolds. Planning and management of water resources has become a complex issue due to inconsistent and conflicting demand of different stakeholders, which are to be met simultaneously through the available source. To control the alarmingly rising complexity as stated above, it is important to implement scientific and well-organized water management approaches to preserve and optimize the existing water resources. There is uncertainty and vagueness in the demand, accessibility of labor in the command area for farming, benefits from crops, and inflows into the reservoir. Thus, it becomes an exceptionally challenging task that includes-: i) scheduling water resources schemes and ii) prioritization of multi-sectored demand.
Fuzzy set theory has been identified as an alternative approach to handle such uncertainty and vagueness in planning of water resource schemes with multiple objectives. The subsequent paragraphs include the applications of MOFLP approaches in integrated management of water resources. The MOFLP approach was applied by Raju and Kumar (2000) in irrigation planning for Sri Ram Irrigation Project in India. The compromised solution for three conflicting objectives was found with degree of truth as 0.69.
A MOFLP approach was developed by Sahoo et al. (2006) for the development and implementation of land-water-crop system of the Mahanadi-Kathajodi Delta in Eastern India. The study model optimizes the financial income, production from crop, and utilization of labour. It also proposed suitable cropping patterns and irrigation intensities for the command area. The study demonstrates and specifies that fuzzy rule-based management methods may be considered to deal with multiple-criteria decision systems.
The MOFLP approach is needed to develop an improved policy for reservoir operation where there is uncertainty in a number of parameters, such as resources, coefficients of objective functions, and technological coefficients, Chaudhari and Anand Raj (2010). The proposed approach is evaluated on a system of four parallel and serially connected reservoirs. Maximization of releases for hydropower and irrigation is considered as the objective function. The study concluded that the fuzzification of the constraints in the MOFLP technique for parallel and series reservoir systems, results in distinct operating strategies and increasing the resilience of the policymakers. Morankar et al. (2013) used MOFLP model for optimal planning of irrigation for Khadakwasla project in India considering three objectives, viz., net irrigation benefit, crop production and labour employment in the command area. The nonlinear, hyperbolic and exponential functions are considered as the membership functions in the model which provides degree of satisfaction as 0.58, 0.79 and 0.49, respectively. Gurav (2010, 2012) had developed optimal irrigation planning models for Jayakwadi Phase-I and Phase-II, respectively in Maharashtra, India. The four objectives-i). Net irrigation benefit (NB), ii). Crop production (YP), iii). Employment generation (EG) and iv). Manure utilization (MU), were considered with crisp constraints in Phase-I. The fuzziness in all the parameters was considered in Phase-II with the same objective functions. The degree of satisfaction (λ = 0.73) obtained for phase-II was found significantly better than Phase-I (λ = 0.58). Raju et al. (2012) applied multi-objective Differential Evolution (MODE) in irrigation planning and its application is demonstrated through a case study of Mahi Bajaj Sagar Project, Rajasthan, India. Three conflicting objectives, namely net benefits, agricultural production and labour employment, were considered for the MODE; and non-dominated alternative was generated using K means clustering for effective decision in the irrigation planning. Mirajkar and Patel (2016) had identified optimal operational strategies for the Ukai reservoir in India. These strategies were developed to maximize net irrigation benefits and employment generation, municipal and industrial revenues and to minimize cultivation costs. To optimize the objective functions, maximum-minimum operator (max-min), two-phase MOFLP (TPMOFLP), and fuzzy compromise technique (average operator) are employed as MOFLP models. The average operator is recommended for the study area vis-à-vis other MOFLP models. Arunkumar and Jothiprakash (2016) proposed the MOFLP solution for the case study of Kukadi irrigation project, Maharashtra, India for objectives like net irrigation profit and crop production. The MOFLP model resulted to a level of satisfaction of 0.46, wherein the canals system performed well with the reliability exceeding 95%. The MOFLP approach was used for optimizing water and land resources under uncertainty, Ren et al. (2017). The MOFLP model evaluated three objectives, namely, minimization of irrigation water by reducing the irrigation areas with guaranteed food security, maximize crop production, and minimize the groundwater exploitation. The study demonstrated that proposed strategy optimizes water for irrigation and land use uncertainty.
The concept of intuitionistic fuzzy set (IFS) was introduced by Atanassov (1986). The properties and an example of the intuitionistic fuzzy set were given to prove the generalization of the concept of 'fuzzy set'. Further, Angelov (1997) found that nonmembership or degree of rejection is not complementary to membership (degree of acceptance) of objective functions. The study found that Intuitionistic fuzzy optimization i.e., IFO solutions provide better solutions to the objectives vis-à-vis classical fuzzy and crisp optimization techniques.
Hernandez and Uddameri (2010) applied a multi-criteria decision-making method based on Atanassov's intuitionistic fuzzy sets (A-IFS) concept for ranking the best management exercises in an agricultural field in the region of South Texas of the USA. The solution developed in the ranking of the different alternatives wherein 'irrigation scheduling' and 'brush control' were found to be the most favored and least favored alternatives, respectively. Hashemi et al. (2013), based on Atanassov's intuitionistic fuzzy sets concept, introduced an innovative compromise ratio method for multiple criteria decision-making in the management of water resources. The concept of hesitation index is introduced by Garai and Roy (2013) for optimization of a hypothetical mathematical problem wherein the maximization of the degree of acceptance and minimization of the degree of rejection, and hesitation were considered as the objective functions. Bharati and Singh (2014) have described multi-objective linear programming using IFO and compared linear membership and non-linear membership functions and their impacts on optimization problem. The IFO approach using non-linear membership and non-linear nonmembership functions, demonstrated better performance over linear membership and non-membership functions. Li et al. (2017) demonstrated the intuitionistic fuzzy multi-objective non-linear programming approach for the distribution of water for irrigation under dry and wet conditions for reducing the water scarcity, and obtain the solutions for crop yield, water saving, and cost reduction to get a balanced plan for water allocation systems. Jafarian et al. (2018) suggested intuitionistic fuzzy multi-objective geometric programming to solve multi-objective nonlinear programming problems. The proposed method was demonstrated through a numerical example with three-objective functions and provides the best compromised solution of 1 3 objective functions for different approaches together with membership and non-membership function values. Pawar et al. (2022a) used the IFO approach as new optimization method to develop the cropping pattern in the Kakarapar Right Bank Main Canal command area of the Ukai-Kakrapar water resources project in India. Apart from the optimal cropping pattern, additional uncertainty parameters like degree of acceptance, rejection, and hesitation were also proposed. Pawar et al. (2022b) used IFO MOFLP model to optimize crop allocation in the Ukai-Kakrapar water resources project with three objectives, namely, maximization of Net irrigation benefits (NIB), minimize of cost of cultivation (CC) and maximization of municipal and industrial revenues (MI). TPIFO MOFLP model provided better results in terms of irrigation intensity, objective function values, degree of acceptance, degree of rejection and hesitation index than other IFO MOFLP models.
The previous studies on MOFLP approaches considered the degree of satisfaction as the only performance parameter for optimization of relevant objective functions of water resources projects. In contrast, the IFO MOFLP approaches, proposed in current study, give two additional parameters, namely, degree of rejection and degree of hesitation, apart from degree of acceptance. This would enable the decision-makers to make the robust decisions in implementing the optimized strategies of water resources projects.
The present study is aimed to address the following research objectives to: (i) Develop multi-objective intuitionistic fuzzy optimization (IFO) approaches to solve an intricate real-world problem of water resources with four conflicting objectives. (ii) Analyze the parameters with sensitivity of performance like degree of acceptance, degree of rejection, and hesitation index with scaling factor that would help the planners to take suitable decisions in implementation of IFO MOFLP solutions. (iii) Compare and demonstrate the merits of the proposed IFO solution with previous multi-objective fuzzy linear programming (MOFLP) solutions for the same study area. (iv) Compare the cropping pattern obtained from IFO MOFLP solution with real cropping pattern to signify the importance of such scientific approaches in planning of water resources.

Study Area
The Ukai-Kakrapar water resources project is the second-largest multi-purpose reservoir in Gujarat, India. Figure 1 shows the index map of the Ukai-Kakrapar water resources project which has three canal command areas in the system. The Ukai left bank main canal (ULBMC), which originates from the Ukai reservoir itself, has the command area of 66,168 hectares. The Kakrapar right bank main canal (KRBMC) and Kakrapar left bank main canal (KLBMC) originate from Kakrapar weir, which is located on the Tapi River at 30 km downstream of Ukai reservoir. The KRBMC and KLBMC encompass the command areas of 1, 13,123 ha and 1, 45,335 ha, respectively. Table 1 includes the planned cropping pattern of each command area. Historical monthly inflows of 36 years into the Ukai reservoir were analyzed (Mirajkar and Patel 2016) and it was reported that the monthly inflow pattern follows the lognormal distribution. The monthly inflows corresponding to 75% probability of exceedance were then computed using the fitted probability distribution. The modified Penman method was adopted for computation of water requirements of the crops wherein due allowance was given to the effective rainfall in the command areas for arriving the net irrigation requirements of different Table 1 Principal Crops in the Ukai Command Areas (Mirajkar and Patel 2016) k, r, hw, and p represent the Kharif, rabi, hot weather, and perennial crops, respectively Juwar/Bajri/other (r) Juwar/Bajri/other (r) Juwar/Bajri (r) 7 Paddy (r) Pulses and other (r) Paddy (hw) 8 Pulses and other (r) Paddy (hw) Groundnuts (hw) 9 Groundnuts (r) Groundnuts (hw) Cotton (ts) 10 Vegetables (hw) Vegetables (hw) Vegetables (ts) 11 Groundnuts (hw) Sugarcane (p) Sugarcane (p) 12 Other (hw) Bananas (p) Bananas (p) 13 Paddy (hw) --14 Bananas and other (p) --15 Sugarcane (p) --

Fig. 1
Index map of the Ukai-Kakrapar water resources project, Gujarat, India crops. Further, the overall irrigation efficiency for the command areas is considered to attain the gross irrigation requirement of the crop.

Methodology and Model Development
The detailed description of each objective and constraint are illustrated in Mirajkar and Patel (2016). However, brief descriptions of objective functions and their related constraints are given in the following paragraphs.

Objective Functions and Constraints
Four objective functions, namely maximization of net irrigation benefits (Z 1 ), employment generation (Z 2 ), minimization of cost of cultivation (Z 3 ), and maximization of revenue generation from municipal and industrial supplies (Z 4 ), were used to obtain the optimal cropping pattern in the command areas of Ukai-Kakrapar Water Resources Project. The individual Linear Programming (LP) solutions of each objective were obtained using the Modified Simplex method in LINGO version 18.0 under relevant constraints, i.e., water allocation constraint, maximum sowing area constraint, socio-economic constraint, canal capacity constraint, reservoir storage capacity constraint, continuity constraint, overflow constraint, and releases to the municipal and industrial supplies and labor constraints. In the current work, a multi-objective IFO model has been developed using individual LP solutions of specified objective functions to establish the cropping pattern in the command areas of Ukai-Kakrapar Water Resources Project.

Intuitionistic Fuzzy Optimization
Intuitionistic fuzzy optimization (IFO) is a recent approach in defining fuzzy sets. The IFO is useful for solving the problems wherein available acquaintance is insufficient and impreciseness is associated with the solutions. In ordinary fuzzy sets, there is only consideration of membership (degree of satisfaction) whereas Intuitionistic fuzzy sets consider both membership (degree of acceptance), non-membership functions (degree of rejection) along with hesitation index. In present study, a multi-purpose Ukai-kakrapar water resources project has been considered for obtaining the optimal cropping pattern and releases from the reservoir. The algorithm for solving IFO MOFLP and its analytical aspects are included in the Supplementary material at Appendix A.
The detailed methodology adopted for formulation of IFO MOFLP, IFO MOFLP with hesitation index and TPIFO MOFLP is included in Fig. 2. The payoff matrix for developing IFO MOFLP model is included in Table 2.

Results and Discussions
The complete analyses of IFO MOFLP, IFO MOFLP with hesitation index, and Two-phase IFO MOFLP, for the chosen study area, i.e., Ukai-Kakrapar water resources project, as per the approach discussed in Sub-Section 3.2 are included in the Supplementary material at Appendix B.
The relative performance of IFO MOFLP models, sensitivity analysis of performance parameters (α, β,and π) with scaling factor, comparison of proposed IFO MOFLP model (TPIFO MOFLP model) with Compromised MOFLP (Average Operator Case-I) model (Mirajkar and Patel 2016), cropping pattern from the proposed TPIFO MOFLP model and rule curve of the reservoir are included in following paragraphs:-  Figs. 3 and 4, it is clearly observed that the α is invariant with change in scaling factors. However, the β decreases and π increases with increase in scaling factor. The planner has to strike the right balance between β and π, depending upon the degrees of rejection and hesitation (uncertainity) desired in the selection of suitable value of S f . The sensitivity analysis of TPIFO MOFLP models could not be accomplished due to availability of limited values of S f for the feasible solutions. In order to obtain the highest irrigation intensity and associated values of objective functions, the best values of S f , for IFO MOFLP, IFO MOFLP with hesitation index, and TPIFO MOFLP, have been selected by striking the right balance between α, β and π (Figs. 3 and 4).   Table 4).

Relative Performance of IFO Models
Similarly, IFO MOFLP model with hesitation index, for S f = 0.27, provided the optimal solutions for each of the four objective functions, i.e., NIB, EG, CC, and MI as Rs. 10581.12 million, 31922.65 thousand workdays, Rs. 5443.17 million, and Rs.1991.69 million ( Table 3). The irrigation intensity for the whole catchment has been estimated to be 102.09%, and corresponding values of ULBMC, KLBMC, and KRBMC are 135.44%, 104.83% and 79.07%, respectively for S f = 0.27 (Table 4).
The results of the objective functions obtained by IFO with hesitation index have been found to improve marginally as compared to the IFO MOFLP model. The solutions of individual objective functions from IFO solutions with hesitation index for S f = 0.27 were substituted in Eqs. (A1), (A3) and (A2), (A4) to get the membership function and nonmembership function values for the maximization and minimization type objective function, respectively. The overall values of α, β and π for S f = 0.27 obtained using Eqs. (A12) are 0.51, 0.25 and 0.24, respectively. The IFO MOFLP model with hesitation index gives additional parameter of hesitation index (π) compared to IFO MOFLP model, wherein the values of objective functions, degree of acceptance and degree of rejection are the same.
The optimal TPIFO solution gives the improvement in results of each the objective functions, i.e., NIB, EG, CC, and MI as Rs.10836.19 million, 34980.4 thousand workdays, Rs.5672.28 million, and Rs. 2314.03 million, respectively. The irrigation intensity obtained from the foregoing model for ULBMC, KLBMC and KRBMC are 132.44%, 116.05% and 95.41%, respectively. Also, the degree of acceptance (α), degree of rejection (β), and degree of hesitation (π) have been estimated as 0.68, 0.19, and 0.13, respectively and these values are significantly better than IFO MOFLP and IFO MOFLP with a hesitation index (See Table 4). Further, the values of objective functions, namely, net irrigation benefits, employment generation, cost of cultivation and revenue generation from municipal and industrial water supply are better than corresponding values of IFO MOFLP and IFO MOFLP with a hesitation index (See Table 4).
From the above discussion, it is also apparent that irrigation intensity for ULBMC is the highest followed by KLBMC and KRBMC sub-command areas. The simulated irrigation intensity for ULBMC, being the lined canal, is higher than KLBMC and KRBMC. Keeping in view, the proficiency of TPIFO MOFLP model, its relative performance is compared with Compromised MOFLP model (Average Operator Case-I) proposed by Mirajkar and Patel (2016) for the same study area.

Performance of TPIFO Model with Compromised MOFLP (Average Operator Case-I)
The IFO models [IFO MOFLP, IFO MOFLP with hesitation index, and TPIFO MOFLP] mentioned in the preceding sections were solved for 75% probable inflow conditions into the Ukai reservoir. The comparison of best performing TPIFO MOFLP model with Compromised MOFLP model (Average Operator-case-I), as proposed by Mirajkar and Patel (2016) is discussed in following paragraphs: -As mentioned earlier, the TPIFO MOFLP model gives the values of individual objective functions, i.e., NIB, EG, CC, and MI, as Rs. 10836.19 million, 34980.4 thousand workdays, 5672.28 million Rs., and 2314.03 million Rs., respectively for the UKai-Kakrapar command area with uncertainty parameters like degree of acceptance (α), degree of rejection (β), and degree of hesitation index (π) as 0.68, 0.19, and 0.13. Additionally, the implementation of  Table 3 of Mirajkar and Patel 2016) for 75% probable inflow condition.
The Compromised MOFLP (Average Operator Case-I), though gives marginally better results in terms of some of the objective functions, however, it does not give any uncertainty measures, like degree of rejection (β), and degree of hesitation index (π), in the optimal solutions for taking the robust decision by the decision-makers. Thus, TPIFO MOFLP models can be recommended for obtaining the optimal solutions of a water resources system with the inclusion of degree of acceptance, degree of rejection, and hesitation indices.  Table 5, it is seen that TPIFO MOFLP model invariably gives lesser irrigation intensity in the ULBMC subcommand area as compared to the other IFO MOFLP models and Compromised MOFLP model (Average Operator Case-I) due to lesser area suggested in the former model for the sugarcane crop. Such decrease in the area of sugarcane from the model is partially compensated by allocation of larger areas for juwar/bajri/other rabi crops, pulses/other similar rabi crops, and groundnuts crops in the TPIFO approach. For the KLBMC subcommand area, the TPIFO MOFLP approach gives higher intensity (ref. Table 5) vis-à-vis other IFO MOFLP models and Compromised MOFLP (Average Operator Case-I) due to the allocation of larger areas for juwar/bajri/other rabi crops, groundnuts and hot weather vegetables. Similarly, the TPIFO model gives higher irrigation intensity in Table 5 Areas allocated to different crops (in ha) by IFO MOFLP models for ULBMC, KLBMC and KRBMC compared with Average Operator Case-I under inflows having 75% Probable of Exceedance the KRBMC command area (95.41%) vis-à-vis other IFO MOFLP models due to higher cropped area for juwar/bajri (in rabi season), groundnut (during hot weather), cotton and banana. The increase in the areas of cash crops like cotton, bananas in the command areas due to implementation of TPIFO model, is likely to improve the prosperity of the region. On the other hand, the increase the cop areas of juwar/bajri/other rabi crops, pulses/other similar rabi crops, and groundnuts crops, would help the local population for their selfsustenance in the command area. Table 6 includes the optimum reservoir releases from the proposed TPIFO MOFLP model for ULBMC, KLBMC, and KRBMC subcommand areas to meet the needs of irrigation, municipal supplies, and industrial supplies. Invariably, the optimal irrigation releases from the proposed model are very meager during the Monsoon (July-August and September) months due to heavy rainfall in the command area. While observing the reservoir levels derived from the TPIFO MOFLP model (Fig. 5), it is seen that reservoir is at 96.9 m (in October month) for 75% dependable inflow condition at the end of the monsoon period. Further, it is seen that reservoir level would be at 83.0 m (June) level just before the start of the monsoon which is above the minimum drawdown level of (82.3 m) of the reservoir. On the other hand, the optimal releases are on higher side for the October, January, February, March and April months. Preparations of land and Kor watering for the Rabi crops create extensive need of irrigation in the October month.

Proposed Rule Curve and Release Pattern in the Command Area
Similarly, the water release for January and February months is higher due to the peak demand of Rabi crops during their grown-up stage. Also, the maximum release of water during March and April months are due to water requirement for land preparation for the hot weather crops, their kor watering and accounting for the high evapo-transpiration losses during these months. Further, the releases from the Compromised MOFLP (Average Operator -Case I) for different subcommand areas are lesser than those obtained TPIFO MOFLP model. From Table 6, it can be seen that total release obtained from the proposed TPIFO MOFLP model is 6.62% higher than those obtained from the Average Operator Case-I for the Ukai-kakrapar command area. Such higher releases of water from the TPIFO MOFLP model are due to increase in the additional areas for the groundnut, juwar/ bajri/pulses and other rabi crops, hot weather and cotton crops. The irrigation authorities would be able to plan out the regulation of water levels in the reservoir and release enough water in each sub-command area of the Ukai reservoir with the help of the estimated reservoir levels and optimal releases from the Ukai reservoir (Table 6 and Fig. 5).

Optimized Crop Area in Proposed Model vis-à-vis Actual Crop Area
The relative importance of suggested TPIFO model is emphasized by comparing the optimized crop areas in the current study with the actual crop areas in the command area of the Ukai-Kakrapar water resources project. The 75% probable inflow condition on annual scale was obtained by using the historical data of 36 years which corresponds to the real inflow into the reservoir for the year 2008-09. The cropping pattern obtained from TPIFO model for 75% probable inflow condition is compared with the real cropping pattern for the ULBMC and KLBMC sub-command areas (See Table 7). Such comparison could not be performed for the KRBMC command area due to nonavailability of real cropping pattern for the KRBMC sub-command area. The actual cropping patterns in ULBMC and KLBMC are different than those recommended by the TPIFO model due to non-availability of such scientific procedure for allocating the crops with the irrigation authorities (see Table 7). The following are the added advantages of using the TPIFO model for allocating the crop areas in the command area vis-à-vis the current practices being adopted in the agricultural fields: (i) Irrigation intensity obtained by the proposed model is 18.51% and 56.11% higher for ULBMC and KLBMC respectively (ii) For ULBMC, it is observed that NIB and EG derived from the proposed TPIFO model are 4.71% and 36.12%, respectively higher than actual pattern (iii) For KLBMC, it is observed that NIB and EG derived from the proposed TPIFO model are 61.2% and 102.28%, respectively higher than NIB and EG values estimated from the actual cropping pattern for the years 2008-09 This increase in irrigation intensity from 114.32% to 132.4% (59.94% to 116.05%) leads to higher values of NIB and EG values in the ULBMC (KLBMC) and command areas.

Performance Assessment of the Water Resource System
In order to assess the performance of reservoir system for the proposed TPIFO model, 36 years of historical inflow data  and 100 years (2010-2019) of artificially generated inflow data were utilized for the Ukai reservoir. With the help of Monte Carlo simulations, 100 years of monthly inflow data were generated. The statistical properties like mean and standard deviation for artificially generated data sets were identical to those of the historical data. The best cropping pattern and reference releases were chosen for the inflows with 75% probability of exceedance as derived from the TPIFO MOFLP model. The LINGO 18.0 (extended version) was used for the simulations of releases with historical/artificially generated data for the best cropping pattern obtained from the model. The simulated reservoir releases were compared with those obtained from the recommended TPIFO MOFLP model.  A particular month is considered deficit, if the simulated release for that month is lower than the release recommended from the TPIFO model for the same month. The performance indices, namely, monthly frequency of irrigation deficit (MFID), annual frequency of irrigation deficit (AFID), annual average irrigation deficit (AAID), and percentage annual irrigation deficit (PAID), were used to assess the performance of the reservoir with reference to the releases obtained from TP IFO model. The detailed description of such indices is available in Mirajkar and Patel (2016). Table 8 shows the performance of the reservoir system for historical inflows of 36 years and synthetically generated data sets of 100 years. According to above analyses, the irrigation deficit would rise from 25.92 Mm 3 for the past 36 years to 29.89 Mm 3 during the next 100 years. The monthly frequency irrigation deficit (MFID) (annual frequency irrigation deficit -AFID) would rise from 7.64% (16.67%) for historical inflows to 24.17% (84%) for the synthetically generated data of next 100 years. For historical and synthetically generated data sets, the percentage annual irrigation deficits are 33.56% and 55.06%, respectively. For both the historical and synthetically generated data sets, the simulation yielded the highest deficit during March, April, and May months.

Conclusions
The socio-economic issues of Ukai-Kakrapar water resources system in India have been addressed using the intuitionistic fuzzy optimization (IFO) approach. In comparison to the earlier model suggested by Mirajkar and Patel (2016), the TPIFO MOFLP recommended in present study, provides additional decision-making parameters such as α, β, and π for the same command area. The key conclusions obtained from the foregoing study are included in terms of relative performance of IFO MOFLP models; proposed TPIFO MOFLP model vis-à-vis compromised MOFLP (Average Operator-I) approach, cropping pattern from the proposed IFO MOFLP approach and release pattern from the reservoir: (a) In present study, three intuitionistic fuzzy optimization-based models have been developed to optimize the four conflicting objective functions. Out of three IFO models, i.e., IFO MOFLP, IFO MOFLP with hesitation index, and TPIFO MOFLP, the TPIFO model is proposed for the whole command area of the project due to a higher degree of acceptance, lower values of degree of rejection, and hesitation index, higher irrigation intensity, and better values of the objective functions. (b) The sensitivity analysis of uncertainty parameters with scaling factors indicated that the degree of acceptance (α) is invariant with scaling factors while the degree of rejection (β) decreases and hesitation index (π) increases with scaling factors. Further, it is noticed that optimal solutions of IFO MOFLP and IFO MOFLP with hesitation index are obtained corresponding to the point of intersections of degree of rejection (β) line and degree of hesitation (π) lines for respective models. (c) Further, it is revealed that suggested TPIFO MOFLP model is more efficient than MOFLP model as recommended by Mirajkar and Patel (2016) for the same system. Former model gives higher irrigation intensity and additional performance measures, viz. α, β, and π, which would help the decision-makers to decide its implementation in a particular water resources system.  Table 6 and Fig. 5) with irrigation intensity of 132.44%, 116.05%, and 95.41% respectively. (f) The levels of Ukai reservoir for various months (Fig. 5) and the releases for the subcommand areas throughout various months (Table 6) would help the decision-maker in the field in regulating the reservoir releases to achieve the intended objectives in the command area. (g) The Ukai reservoir has been simulated for historical and synthetically generated inflows for the cropping pattern obtained from the recommended TPIFO MOFLP model for 75% probable inflow conditions. The MFID (AFID) of 7.64% (16.67%) and 24.17% (84%) were found for historical and synthetically generated inflows respectively. Similarly, the AAID (PAID) for historical and synthetically generated inflows were found to be 25.92 Mm 3 (33.56%) and 29.86 Mm 3 (55.06%) respectively. These irrigation deficits were invariably observed in February, March, April, and May months due to severe reduction of flows in these months. (h) The recommended cropping pattern from the TPIFO model is compared with actual cropping for 75% dependable flow condition in the command area. The significant improvements in net irrigation benefit and employment generation are reported due to the implementation of the proposed model.
The methodology presented in the current investigation is generic in nature; the same can be applied to other water resources systems to develop IFO models for management of water resource systems.
The current study has been developed from the accessible data of the year 1975-2010, which is an extension of the Compromised MOFLP (Average Operator Case-I) model recommended by Mirajkar and Patel (2016). The same can be updated by incorporating additional data of latest years for the system in the future.