East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

ARTICLE

ARTICLE INFO

Volume 7(1), 2026

https:

//dx.doi.org/10.4314/eajbcs.v7i1.5S

Intuitionistic Fuzzy Multi-objective

Optimization Method for Determination of

ARTICLE HISTORY

Received: 03 March, 2026

Accepted: 04 June June, 2026

Published Online: 10 June, 2025

Optimal Cropping Pattern

Habtamu Tsegaye Teferi ^1,∗

�

CITATION

Teferi H.T (2026). Intuitionistic Fuzzy

Multi-objective Optimization Method for

Determination of Optimal Cropping

Pattern. East African Journal of

¹Department of Mathematics, College of Natural and Computational Sciences, Wolkite University, Ethiopia

^∗Corresponding author: hbtmtsgy@gmail.com

Biophysical and Computational

Sciences Volume 7(1), 2026. .https://dx.

doi.org/10.4314/eajbcs.v7i1.5S.43-63

Abstract

OPEN ACCESS

Agriculture has become a difficult occupation due to inadequate farming resources and cultivation risks.

Thus, proficient utilization of resources alongside risk-alleviation strategies is essential aspect to realize

sustainable farm benefits. Most of the earlier studies have reported the capability of Operations research

in solving agricultural problems and enhancing farm productivity. However, they have not addressed

effectively the distinctive nature of decision-makers, uncertainties, and associated risks of agriculture. This

study mainly aims to fill these lacunae by applying an intuitionistic fuzzy optimization method to determine

optimal cropping pattern that maximizes overall net benefits, minimizing cultivation costs and workforce

concurrently with regard to procurable agricultural resources. For this purpose, an effective multi-objective

optimization method is formulated, and its effectiveness is verified through proof and numerical example.

The comparison between existing and proposed cropping patterns showed that the proposed patterns

offer several advantages in enhancing overall agricultural benefits sustainably for farmers in the study

area.

This work is licensed under the Creative

Commons open access license (CC

BY-NC 4.0).

East African Journal of Biophysical and

Computational Sciences (EAJBCS) is

already indexed on known databases

like AJOL, DOAJ, CABI ABSTRACTS and

FAO AGRIS.

Keywords: Multi-objective optimization; Intuitionistic fuzzy optimization; Agricultural production

planning; Cropping pattern.

models to support farm activities and contribute to feeding the steadily

growing population (Carravilla & Oliveira, 2013; Weintraub & Romero,

1 Introduction

Today, the burgeoning world’s population increases the demand for

agricultural products and this in turn increasing pressure on the resources

required for production (Wang, 2022). However, to meet the ever-escalating

demand for nourishment, crop production must be boosted either by

increasing land area for cultivation or by enhancing production per unit

area of land (FAO, 2017; Mirkarimi et al., 2013). Since agricultural

resources for farming are very limited all over the world (Guo et al., 2021),

increasing land area for crop production regardless of limiting factors

causes deterioration of available resources. Moreover, climate change,

drought, political disputes, and disease are holding back agricultural

advancement and remain key determinants of food security and poverty

alleviation (Luo et al., 2023; Zerssa et al., 2021). Therefore, it is crucial

to design means of eﬃcient utilization of resources and risk alleviation

strategies to improve the overall agricultural returns sustainably.

2006).

Agriculture is the primary sector of the Ethiopian economy, employing

approximately 85% of the country’s population as workers (Zerssa et al.,

2021). It contributes 50% of Ethiopia’s gross domestic product and earns

over 90% of the foreign exchange (Haile & Kasa, 2015; Zerssa et al., 2021).

The country’s goal for achieving overall economic growth mainly depends

on the accomplishment of agriculture sector (Haile & Kasa, 2015).

Despite the country’s vast irrigable land and water resources, farming is

weather dependent and the production of which depends heavily on the

availability of rainfall (Awulachew & Ayana, 2011). Consequently, most

farmers are exposed to inconsistent rainfall patterns and weather conditions

(Zerssa et al., 2021). Even though Ethiopia is vulnerable to the vagaries of

natural weather conditions, it has substantial agricultural potential because

of its vast areas of fertile land, enormous labor, abundant water resources,

and diverse climate conditions (Awulachew & Ayana, 2011; Kelbore, 2014).

In this regard, the decision of agricultural stakeholders has its own impact

on the achievement of the desired objectives. Even if decision-making for

the optimum utilization of resources is a challenging task for farmers and

agricultural managers, it can be scientiﬁcally addressed using optimization

Due to seasonal variation, Ethiopia has three distinct seasons: Kiremt

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

(June – September), Bega (October – January), and Belg (February – May)

(Gebremichael et al., 2014). Kiremt is the main rainy season, accounting for

about 90% of agricultural production, while Bega is dry and Belg is a short

rainy season contributing the remaining 10% of production (Kelbore, 2014).

literature is reviewed. The optimization method for addressing the practical

problem is outlined in Section 3. A comparative analysis of the proposed

method is presented in Section 4. Section 5 provides the model application

in three subsections: study area description, data collection and analysis,

and problem formulation. A detailed analysis of the results is given in

Section 6. Section 7 summarizes the ﬁndings, discusses limitations, and

provides recommendations for future research.

A cropping pattern is the proportion of land area under various crops that

changes over space and time (G. Singh, 2012). It is the annual succession of

diﬀerent crops and fallow in a particular region and can be reported at the

farm level to address the collective issues of a farming system (Andrews &

Kassam, 1976).

2 Literature Review

Agriculture is one of the ﬁelds where Operations research (OR) models were

ﬁrst employed and they have been extensively applied (Rădulescu et al.,

2014). The capability of addressing MOO problems for decision-making

helps out OR to play a fundamental role in agriculture (Rădulescu et al.,

2014; Weintraub & Romero, 2006).

The principal objective of optimal cropping pattern (OCP) is to identify

the combination of several crops to be cultivated which maximizes the net

return of farming by managing agricultural risks with respect to available

resources (Ouda et al., 2017).

Carravilla and Oliveira (2013) reviewed studies demonstrating the

applications of OR in agriculture at farm and sector levels. Depending

on problem complexity, some used linear programming (LP), while others

employed multi-objective and fuzzy optimization models to address APP

problems.

As the population grows, agricultural resources have been decreasing

and the situation worsens with the spread of drought (Carravilla &

Oliveira, 2013). These circumstances continue to generate many agricultural

inquiries in search for more productive alternatives on a given land area

for the optimal utilization of other agricultural resources (Paudel, 2016).

Therefore, OCP is one of the important and feasible mechanisms to increase

productivity with other integrated scientiﬁc agricultural practices (Luo

et al., 2023; Ouda et al., 2017).

Weintraub and Romero (2006) demonstrated the potential of OR in the

management of agricultural resources and forestry, and their advantages

in simplifying DM in farming activities. In their article, the applications of

OR in APP problems at the farm and regional levels were comprehensively

reviewed. They address uncertainties, risks, environmental conservation,

and discuss future research directions in these areas.

Determining the OCP remains a complex task for farmers and managers

(Pawar et al., 2026), as decisions on machinery selection, input use,

operation timing, and cultivation practices must be made in each cropping

season (Duan et al., 2021).

Environmental, social, and economic factors make agricultural data

inherently uncertain (Bairwa et al., 2013), leading to decision-making under

ill-deﬁned objectives and constraints (Li et al., 2017). Accordingly, risks in

agriculture are better represented with fuzzy numbers than crisp values,

prompting the development and application of fuzzy multi-objective

optimization (FMOO) methods to handle APP problems.

In Ethiopia, where most farmers practice traditional agriculture, these

decisions are largely based on experience, ﬂuctuating market prices,

and other operational factors (Kelbore, 2014). Conversely, policies and

guidelines delivered through extension services often fail to account for

multiple agricultural objectives and constraints, such as weather variability

and production risk. Therefore, more eﬀective strategies are required

through the use of optimization models.

Although many researchers employed FMOO approaches to deal with APP

problems (Amini, 2015; Biswas & Pal, 2005; Gupta et al., 2000; Mirajkar &

Patel, 2012; Mirkarimi et al., 2013; Rasikh et al., 2024; Wang, 2022; X. Zeng

et al., 2010) their studies were limited in scope and did not represent the

real nature of the problems very well (Mahapatra & Roy, 2009). This

is due to insuﬃcient information, incomplete attributes, ill-deﬁnedness,

uncertainties, and vagueness in every aspect of MOO problems (Sen et al.,

2018). Consequently, diﬀerent advanced generalizations are ascertained.

From that, IFS, which is originated by Atanassov (1986), is an eﬀective

generalization of fuzzy sets. It has been used in a wide range of operations

because of its ability to address uncertainties and vagueness in practical

problems. Thus, intuitionistic fuzzy optimization (IFO) (Angelov, 1995) was

introduced to handle diﬀerent pragmatic problems.

The problem of optimizing multiple goals simultaneously under given

constraints is called multi-objective optimization (MOO). It involves

concurrent optimization of incommensurate and conﬂicting goals subject

to diﬀerent constraints (V. Singh & Yadav, 2018).

Many real-life problems are inherently characterized by multiple and

conﬂicting goals with uncertainties (Gupta et al., 2000). So, it is diﬃcult

to deal with such problems using classical optimization techniques.

Hence, intuitionistic fuzzy sets (IFSs) can be used to represent insuﬃcient

information, imprecise concept, uncertainties and the diverse perspectives

of decision-makers (DMs) in a more generalized way compared to fuzzy

sets (Roszkowska et al., 2024). Thus, intuitionistic fuzzy modelling is more

relevant than other classical optimization methods (Pawar et al., 2026).

IFO technique is a relatively recent research ﬁeld in contrast to fuzzy

optimization approaches (Angelov, 1995). It enhances understanding of

the addressed problems and provides valuable insights into their nature

(Roszkowska et al., 2024). Moreover, the output of an investigation

employing IFO is a more valuable analytical means for researchers,

practitioners, and experts.

Several researchers have attempted to address agricultural production

planning (APP) problems employing various MOO methods to recommend

an alternative cropping system for improved outcomes (Luo et al., 2023;

Weintraub & Romero, 2006). However, the ambiguity of the parameters

in the problem, inconsistent natural conditions, the distinctive perspective

of DMs, and associated operational risks in agriculture have not been well

addressed in their studies.

In APP problem, interactions among various natural entities and correlated

factors complicate the management process. These factors enforce DMs to

use advanced optimization models for proﬁcient usage of resources and to

gain better overall beneﬁts.

To overcome the above diﬃculties, it is imperative to formulate an eﬀective

optimization model and design OCP at the farm level by considering the

available resources to assist sustainable crop production.

This study aims to investigate the application of an intuitionistic fuzzy

multi-objective optimization (IFMOO) model and to propose OCP for the

study area that improves farmers’ welfare concerning scarce resources and

reduces various challenges of farming.

Nishad and Singh (2015) employed an intuitionistic fuzzy goal

programming to resolve the land use planning problem. They considered

the agricultural system undertaken by Biswas and Pal, 2005 in which several

seasonal crops were cultivated in a year and diﬀerent productive resources

are taken into account in the model. Their study revealed that IFO method

gives better results in all aspects compared to the results obtained by the

fuzzy optimization method.

The rest of this paper is organized as follows. In Section 2, the related

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Li et al. (2017) formulated an IFMOO model that incorporates MOO,

nonlinear programming (NLP) and intuitionistic fuzzy number (IFN) to

deal with the uncertainties of conﬂicting targets in irrigated agriculture to

support sustainable farming. They employed an IFMOO model to allocate

limited accessible water to rice during growth stages to maximize crop yield,

minimize utilized water and cost of water supply in Heping irrigation area

of northeast China.

3 IFO Method

3.1 Preliminaries

˜

Deﬁnition 3.1.1 (Atanassov, 1986). An IFS ꢀ in the universe - is given by

˜

ꢀ = {hG, ꢀ (G), ꢁ (G)i:G ∈ -}, where ꢀ (G), ꢁ (G), ꢂ (G):

˜

ꢀ

˜

ꢀ

˜

ꢀ

- → [0, 1] such that

0 ≤ ꢀ (G) + ꢁ (G) ≤ 1 and ꢂ (G) = 1 − (ꢀ (G) + ꢁ (G)),

˜

ꢀ

˜

ꢀ

∈

ꢀ

∀G

-. The values ꢀ^ꢀ(G), ꢁ (G) and ꢂ (G) describe the degree

˜

ꢀ

˜

ꢀ

Li et al. (2019) used a multi-objective NLP model in an intuitionistic fuzzy

environment (IFE) for the management of the water-energy-food nexus in

irrigational agriculture, considering the cropping system of the Heihe River

basin in northwest China. In their study, both optimistic and pessimistic

views of DMs were considered under diﬀerent scenarios to maximize

system proﬁt and minimize carbon dioxide emissions, subject to water,

energy, land, and other input resources for cultivating wheat, corn, and

vegetables in the three regions of the basin.

˜

of membership, non-membership, and indeterminacy of G being in ꢀ,

respectively.

˜

Deﬁnition 3.1.2 (Mahapatra & Roy, 2009). A triangular IFN ꢀ is an IFS,

represented as ꢀ = h0 , 0 , 0 ; 0 , 0 , 0 i, where 0 ≤ 0 ≤ 0 ≤ 0 ≤ 0₃⁰,

0

˜

1

2

3

2

1

2

3

1

3

1

with

G−0

1

−G







,

if 0 < G ≤ 0

1

2

3

0

−0

2

1

0

3

ꢀ (G) =

if 0 ≤ G < 0

˜

2

ꢀ

0

−0

2

0³,

otherwise,





Pawar et al., 2022 applied an IFMOO method to determine the OCP of the

Ukai irrigation area in India. Their approach combined minimizing the

aggregated hesitation level of objectives with maximizing the minimum

membership degree and minimizing the maximum non-membership

and

0

−G

2







,

if 0₁⁰< G ≤ 0

0

2

0

−0

2

1

degree.

This produced an OCP that maximized net returns and

G−0

if 0 ≤ G < 0⁰

2

ꢁ (G) =

˜

ꢀ

0

2

3

employment, while minimizing farming expenditures under constraints of

arable land and irrigation water.

0

−0

2



1³,

otherwise.





Li et al., 2020 developed an optimization model for sustainable irrigated

agriculture combining IFMOO, nonlinear mixed-integer, and fuzzy

credibility-constrained programming. Their study aimed to allocate water

and farmland to crops across subareas and seasons to optimize net returns

while considering socioeconomic and ecological objectives. The algorithm

was successfully applied to crop planning in the Heping irrigation area of

China.

Deﬁnition 3.1.3 (S. K. Singh & Yadav, 2015a).

0

˜

Let ꢀ = h0 , 0 , 0 ; 0 , 0 , 0 i be a triangular IFN, the accuracy function of

1

2

3

2

1

3

0

+0 +40 +0 +0

3

1

2

3

1

˜

ꢀ is denoted by Γ(ꢀ) and deﬁned as Γ(ꢀ) =

Accuracy function Γ is used to defuzzify IFNs.

.

8

In handling of practical problems, vagueness and uncertainty can be

addressed in an IFE by considering parameters as IFNs and treating

inequality and equality as intuitionistic fuzzy inequalities and equality.

Based on this principle, an IFMOO problem is formulated (V. Singh & Yadav,

2018)

Kousar et al., 2022 formulated an IFO method considering all parameters

and variables as IFN to optimize the production of ﬁve types of fruits

in Baluchistan region, Pakistan.Their results showed that the fully IFO

technique has an imperative advantage to consider the ﬂuctuating nature

of prices and input resources more eﬃciently.

e

f

max { 5 (-), 5 (-), ..., 5 ₁(-)},

1

2

:

e

min { 5_{: +1}(-), 5_{: +2}(-), ..., 5_:(-)}

1

L. Zeng et al. (2020) studied sustainable resource management for the

Zhanghe Reservoir irrigation in central China using an interval stochastic

multi-objective mixed-integer model in an IFE. Their objectives were to

maximize crop production, hydroelectricity, and water allocation while

optimizing cropland under constraints of water availability, crop demand,

land policy, and electricity generation. The results demonstrated eﬃcient

farmland and water management to support food security and mitigate

global warming sustainably.

subject to

(1)

,e(-) - e2₈, 8 = 1, 2, . . . , < ,

,e(-) ¥ e2 , 8 = <₁+ 1, <₁¹+ 2, . . . , < ,

8

2

8

,e(-) ≈ e2₈, 8 = <₂+ 1, <₂+ 2, . . . , <,

8

- ≥ 0,

e

where 5₉(-) and ,e(-) are intuitionistic fuzzy functions, - is n-dimensional

8

A few researchers have used IFMOO models to handle uncertainties and

risks in APP problems. As noted above, investigators have applied various

IFMOO techniques to suggest alternative cropping patterns for better

outcomes. However, they have not eﬀectively addressed diﬀerences among

DMs, uncertainties, and risks that severely aﬀect crop production.

For instance, in most studies, parameters are not consistently considered

as IFNs, constraints are not incorporated into the solution framework,

DM preferences and stakeholder interactions are overlooked, and the

sustainability of objectives is inadequately addressed.

variable ∀9 = 1, 2, ..., : and ∀8 = 1, 2, ..., <.

To reformulate problem (1) as an equivalent crisp MOO problem, each

parameter has to be defuzziﬁed applying the accuracy function (Rădulescu

et al., 2014). Then, the degrees of acceptance and rejection of the objectives

have to be described to form a single objective optimization problem.

Accordingly, the solution of the IFMOO problem can be found by solving

such an equivalent single objective problem (S. K. Singh & Yadav, 2015b).

The IFMOO problem (1) can be changed into the following equivalent

deterministic MOO problem (V. Singh & Yadav, 2018):

To overcome these shortcomings, the present study formulates an IFMOO

model to design an OCP that maximizes sustainable net beneﬁts for a

large-scale farm (LSF) and farmers in Gefersa kebele. It also aids farmers,

development agents, and extension workers in complex decision-making

under multiple perspectives and scenarios for a LSF in Abeshge district.

max { 5 (-), 5 (-), ..., 5 (-)},

1

2

:

min { 5_{: +1}(-), 5_{: +2}(-)¹, ..., 5_:(-)}

1

subject to

,₈(-) ≤ 2₈, 8 = 1, 2, . . . , < ,

(2)

, (-) ≥ 2 , 8 = <₁+ 1, <₁¹+ 2, . . . , < ,

2

8

,₈(-) = 2₈, 8 = <₂+ 1, <₂+ 2, ..., <,

- ≥ 0,

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

where 5₉(-) and ,₈(-) are real-valued functions and 2₈∈ R,

∀9 = 1, 2, ..., : and ∀8 = 1, 2, ..., <.

of the maximization problem are respectively described as follows:



0,

if 5₉(-) ≤ !₉



Deﬁnition 3.1.4 (Cristofari et al., 2024). Let S be the set of all feasible

solutions of problem (2), and let -, -^∗∈ S. Then, -^∗is said to be a

Pareto optimal solution (POS) for problem (2) if and only if there does





5 (-)−!

9

ꢀ_*( 5₉(-)) =

(4)

,

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,

9

* −!

1,

9



not exist - ∈ S such that 5₉(-^∗) ≤ 5₉(-), ∀9 = 1, 2, . . . , : , and ∃9 ∈



1

{1, 2, . . . , : } such that 5 (-^∗) < 5 (-).

1

9

and



1,

if 5₉(-) ≤ !₉− ꢄ₉

if !₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,

T heorem 3.1.1 (Xu & Cai, 2012; Xu & Yager, 2006).





* − 5 (-)

˜

9

Let ꢁ = (ꢀ , ꢁ ), 9 = 1, 2, ..., : be a collection of intuitionistic fuzzy values,

˜

9

ꢁ

9

ꢁ_*( 5₉(-)) =

,

(5)

9

* −(! −ꢄ )

0,

9



and let F , F , ..., F be the corresponding weights, where F₉∈ (0, 1) and

1

2

:

Í



:

₉₌₁F₉= 1. Then, the aggregated value using the intuitionistic fuzzy

where ꢄ₉is a tolerance value of the 9th objective and deﬁned as ꢄ₉

ꢃ(*₉− ! ) and ꢃ ∈ (0, 1), ∀9 = 1, 2, ..., : . If *₉= !₉, then we deﬁne

=

weighted geometric (IFWG) operator is given by

1

9

ꢀ_*( 5₉(-)) = 1.

9

The respective exponential membership function (ꢀ_*^ꢂ( 5₉(-))) and

:

9

Ö

IFWG (ꢁ , ꢁ , ..., ꢁ ) =

ꢀ^F_˜⁹, 1 −

(1 − ꢁ )

(3)

non-membership function (ꢁ_*^ꢂ( 5₉(-))) for the 9th objective respectively

©

-

ª

®

F

9

˜

F

1

2

˜

:

ꢁ

9

ꢁ

9

9=1

«

¬

deﬁned as



0₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,



3.2 The Solution Method





((* − 5 (-))/(* −! )) −3

9

ꢀ_*^ꢂ( 5₉(-)) =

(6)

4

−4

,

Most existing studies consider the optimistic variant of the problem (Kis

et al., 2021), paying little attention to alternative perspectives in the solution

process. However, this consideration has its own limitations in addressing

practical problems. Due to the inconsistent nature of human judgment,

DMs may deviate from their initial standpoint after evaluating the obtained

solution against the intended goals. Consequently, existing methods

overlook this important aspect of DM judgment and insights. Therefore,

identifying the inﬂuence of optimistic, pessimistic, and mixed perspectives

is highly valuable for obtaining consistent and robust solutions to MOO

problems (Chen et al., 2023; Mahajan & Gupta, 2021b). The main diﬀerence

among these three perspectives arises from the choice of violation and

tolerance values used to determine the non-membership degrees, while the

membership function remains identical across all cases.

−3

9



1−4



1,



and





1₋, ₃

if 5₉(-) ≤ !₉− ꢄ₉

if !₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉.





(( 5 (-)+ꢄ −! )/(* +ꢄ −! )) −3

9

ꢁ_*^ꢂ( 5₉(-)) =

4

−4

,

−3

9



1−4



0,



(7)

where, 3₉is the shape parameter.

The linear membership and non-membership functions for minimization

objectives can be described as

In an IFE, parameter values, aspiration levels, coeﬃcients of the objectives

and constraints, as well as the equalities and inequalities in the model, are

ﬂexible. Consequently, the maximum and minimum allowable values of the

constraints and objectives can vary.



1,

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,





* − 5 (-)

9

ꢀ_!( 5₉(-)) =

(8)

,

9

* −!

0,

9





To incorporate ﬂexibility in the constraints, a violation parameter ℓ₈is

and

associated with the 8th constraint, 8 = 1, 2, . . . , < , as speciﬁed by the DMs.

1

For constraints of the form ≤, the upper bound 2₈is relaxed to 2₈+ ꢃ(ℓ₈) in



0,

if 5₉(-) ≤ !₉





the solution procedure (Tsegaye et al., 2021).

5 (-)−!

9

ꢁ_!( 5₉(-)) =

,

if !₉< 5₉(-) < *₉+ ꢄ₉

(9)

9

(* +ꢄ )−!

1,

9



if 5₉(-) ≥ *₉+ ꢄ₉,

The membership and non-membership functions for each objective are

described based on the diﬀerence between the maximum *₉and minimum

!₉achievable goals, which are identiﬁed using a table of extreme solutions.

Then, the tolerance variables ꢄ₉and ꢅ₉are obtained by using



where ꢄ₉is a tolerance value of the 9th objective and deﬁned as ꢄ₉

=

ꢃ(* − ! ) and ꢃ ∈ (0, 1), ∀9 = : + 1, : + 2, ..., :. If *₉= !₉, then we

1

deﬁne ꢀ_!⁹( 5₉(-)) = 1.

9

The corresponding exponential non-membership function (ꢀ^ꢂ_!( 5₉(-))) and

ꢄ₉= ꢃ(*₉− !₉) and ꢅ₉= ꢃ(*₉− (!₉− ꢄ₉)) = ꢄ₉(1 + ꢃ) where

9

!₉= min{ 5₉(-)}, *₉= max{ 5₉(-)}, and ꢃ ∈ (0, 1), 9 = 1, 2, ..., :.

non-membership function (ꢁ_!^ꢂ( 5₉(-))) for the 9th objective respectively

9

deﬁned as

The membership and non-membership functions are generally

characterized by nonlinear behavior due to instantaneous variation at each

solution point. Among nonlinear functions, the exponential function is

preferable for the IFMOO problem because of its eﬃciency and ﬂexibility in

evaluating the marginal values of objectives and constraints (Ahmadini

& Ahmad, 2021; Mahajan & Gupta, 2021a). Accordingly, exponential

membership and non-membership functions are formulated to describe

the optimistic, pessimistic, and mixed features and to obtain an eﬃcient

solution for the IFMOO problem.



0₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,





(( 5 (-)−! )/(* −! )) −3

9

ꢀ^ꢂ_!( 5₉(-)) =

(10)

4

−4

,

−3

9



1−4



1,



and





1₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < *₉+ ꢄ₉

if 5₉(-) ≥ *₉+ ꢄ₉.





((* +ꢄ − 5 (-))/(* +ꢄ −! )) −3

9 9

9

ꢁ_!^ꢂ( 5₉(-)) =

4

−4

,

3.2.1 The optimistic approach

−3

9



1−4



In the optimistic approach, DM takes a liberal view of rejection (V.

0,



Singh et al., 2021). Therefore, the linear membership (ꢀ_*( 5₉(-))) and

(11)

9

non-membership (ꢁ_*( 5₉(-))) functions for the 9th objective function 5₉(-)

Their general shape is shown in Figure 1.

9

Teferi H.T.,(2026)

46

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

(a)

(b)

(a)

(b)

Figure 2: Exponential membership and non-membership functions for maximization

(a) and minimization (b) objectives under a pessimistic approach.

Figure 1: Exponential membership and non-membership functions for maximization

(a) and minimization (b) objectives under the optimistic approach.

3.2.2 The mixed approach

In a mixed approach, the DM is not ﬂexible in rejecting and is not capable

3.2.2 The pessimistic approach

of extra acceptance (V. Singh & Yadav, 2018). The linear non-membership

In the pessimistic approach, the DM is presumably extra cautious about

function ꢁ_*( 5₉(-))) of the 9th objective function 5₉(-) to the maximization

9

acceptance (V. Singh et al., 2021). The linear non-membership function

problem is deﬁned as

ꢁ_*( 5₉(-)) of the 9th objective 5₉(-) under a pessimistic approach to the

9

maximization problem is expressed as



1₍,_{! +ꢅ )− 5 (-)}

if 5₉(-) ≤ !₉− ꢄ₉





9



ꢁ_*( 5₉(-)) =

,

if !₉− ꢄ₉< 5₉(-) < !₉+ ꢅ₉

if 5₉(-) ≥ !₉+ ꢅ₉,

(16)

1,

if 5₉(-) ≤ !₉





9

(! +ꢅ )−(! −ꢄ )

9



(! +ꢄ )− 5 (-)

9

0,

ꢁ_*( 5₉(-)) =

,

if !₉< 5₉(-) < !₉+ ꢄ₉

if 5₉(-) ≥ !₉+ ꢄ₉,

(12)



9

(! +ꢄ )−!

0,

9





where ꢄ₉and ꢅ₉are the tolerance variables of the 9th objective and deﬁned

as ꢄ₉= ꢃ(*₉− !₉),

∀9 = 1, 2, ..., : .

1

ꢅ = ꢄ (1 + ꢃ), ꢃ ∈ (0, 1), 9 = 1, 2, ..., : .

1

9

The corresponding exponential non-membership function ꢁ_*^ꢂ( 5₉(-)) is

9

deﬁned as

constructed as



1₋, ₃

if 5₉(-) ≤ !₉

if !₉< 5₉(-) < !₉+ ꢄ₉

if 5₉(-) ≥ !₉+ ꢄ₉.









1₋, ₃

if 5₉(-) ≤ !₉− ꢄ₉

if !₉− ꢄ₉< 5₉(-) < !₉+ ꢅ₉

if 5₉(-) ≥ !₉+ ꢅ₉.

(( 5 (-)−! )/ꢄ

)

−3

9

ꢁ_*^ꢂ( 5₉(-)) =

(13)

(( 5 (-)−(! −ꢄ ))/(ꢅ +ꢄ )) −3

4

−4





,

9

−3

ꢁ_*^ꢂ( 5₉(-)) =

4

−4

9



1−4

,

−3

9



0,

1−4



0,



(17)

The linear non-membership function ꢁ_!( 5₉(-)) of the 9th objective 5₉(-)

9

The linear non-membership function ꢁ_!( 5₉(-))) of the 9th objective function

9

under a pessimistic approach to the minimization problem is expressed as

5₉(-) to the minimization problem is deﬁned as



0,

if 5₉(-) ≤ *₉− ꢄ₉

if *₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉,







5 (-)−(* −ꢄ )

1,

if 5₉(-) ≤ *₉− ꢅ₉

if *₉− ꢅ₉< 5₉(-) < *₉+ ꢄ₉

if 5₉(-) ≥ *₉+ ꢄ₉,

9





ꢁ_!( 5₉(-)) =

,

(14)

5 (-)−(* −ꢅ )

9

* −(* −ꢄ )

1,

9



ꢁ_!( 5₉(-)) =

,

(18)

9

(* +ꢄ )−(* −ꢅ )

0,

9









∀9 = :₁+ 1, :₁+ 2, ..., :.

The corresponding exponential non-membership function ꢁ_!^ꢂ( 5₉(-)) is

where ꢄ₉and ꢅ₉are the tolerance variables of the 9th objective and deﬁned

9

as ꢄ₉= ꢃ(*₉− !₉), ꢅ₉= ꢄ (1 + ꢃ), ꢃ ∈ (0, 1), 9 = : + 1, : + 2, ..., :.

1

9

deﬁned as

The corresponding exponential non-membership function ꢁ_!^ꢂ( 5₉(-)) is

9



0₋, ₃

if 5₉(-) ≤ *₉− ꢄ₉

if *₉− ꢄ₉< 5₉(-) < *₉

if 5₉(-) ≥ *₉.



constructed as





(* − 5 (-))/ꢄ

)

−3

9

ꢁ_!^ꢂ( 5₉(-)) =

(15)

4

−4

,

−3

9





0₋, ₃

if 5₉(-) ≤ *₉− ꢅ₉

if *₉− ꢅ₉< 5₉(-) < *₉+ ꢄ₉

if 5₉(-) ≥ *₉+ ꢄ₉.

1−4





1,

((* +ꢄ )− 5 (-))/(ꢅ +ꢄ )) −3



9

ꢁ_!^ꢂ( 5₉(-)) =

4

−4

,

−3

9



1−4



1,



Their possible shape is shown in Figure 2.

(19)

Their general shape is shown in Figure 3.

Teferi H.T.,(2026)

47

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Problem (21) can be expressed based on the DM’s perspectives, using

the membership and non-membership functions constructed from Sections

3.2.1 to 3.2.3, as presented below.

From an optimistic viewpoint, problem (21) can be expressed as

:

Ö

F

©

-

ª

®

©

-

ª

®

F

9

max

ꢆ

×

(1 − ꢇ

)

>

9

9=1

«

¬

«

¬

subject to

3

(1−((* − 5 (-))/(* −! )))

3

9

4

+ ꢆ (1 − 4 ) ≥ 1, 9 = 1, 2, ..., :

>

1

9

(1−(( 5 (-)−! )/(* −! )))

3

9

+ ꢆ (1 − 4 ) ≥ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(22)

(23)

(24)

(1−(( 5 (-)−(! −ꢄ ))/(* −(! −ꢄ ))))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = 1, 2, ..., :

>

1

9

(1−(((* +ꢄ )− 5 (-))/((* +ꢄ )−! )))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(a)

(b)

0 ≤ ꢆ + ꢇ ≤ 1, 9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

Figure 3: Exponential membership and non-membership functions for maximization

(a) and minimization (b) objectives under the mixed approach.

0 ≤ ꢇ ≤ ꢆ

,

9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

8

- ≥ 0.

After converting an IFMOO problem into its equivalent crisp MOO

problem using the accuracy function (S. K. Singh & Yadav, 2015a), the

membership and non-membership functions are constructed based on the

DMs’ viewpoints. Accordingly, aggregation operators have been proposed

(Mahajan & Gupta, 2021b; S. K. Singh & Yadav, 2015b; V. Singh &

Yadav, 2018) to combine the membership and non-membership functions.

However, such aggregation methods have certain limitations, as they fail

to consider for both the satisfaction and dissatisfaction associated with

all objectives. To address this limitation, the IFWG operator, previously

applied to various multicriteria decision-making problems, is extended in

this study to the MOO problem.

The pessimistic variant of problem (21) described as

:

Ö

F

©

-

ª

®

©

-

ª

®

F

9

max

ꢆ

×

(1 − ꢇ

)

>

9

9=1

«

¬

«

¬

subject to

3

(1−((* − 5 (-))/(* −! )))

3

9

4

+ ꢆ (1 − 4 ) ≥ 1, 9 = 1, 2, ..., :

>

1

9

To formulate a single aggregation operator based on Theorem 3.1.1, a

multiplicative combination is employed to integrate the independently

deﬁned degrees of acceptance and rejection. This approach helps to

emphasize the interaction between the two degrees. Accordingly, the

aggregation operator can be expressed as

(1−(( 5 (-)−! )/(* −! )))

3

9

+ ꢆ (1 − 4 ) ≥ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(1−(( 5 (-)−! )/ꢄ ))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = 1, 2, ..., :

>

1

9

(1−((* − 5 (-))/ꢄ ))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = : + 1, : + 2, ..., :

>

1

9

0 ≤ ꢆ + ꢇ ≤ 1, 9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

:

1

9

Ö

©

-

ª

©

ª

®

9

F

9

/(ꢀ, ꢁ) =

ꢀ^F₉× 1 − (1 −

(1 − ꢁ₉)

)

®

-

0 ≤ ꢇ ≤ ꢆ

,

9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

9=1

«

¬

«

¬

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

8

(20)

:

- ≥ 0.

Ö

ꢀ^F₉

×

(1 − ꢁ₉)

©

-

ª

®

©

-

ª

9

F

9

=

®

9=1

«

¬

«

¬

For the mixed approach, problem (21) is expressed as

Therefore, the IFMOO problem (1) can be solved using an equivalent crisp

model by employing the aggregation operator (20) as follows:

:

Ö

©

-

ª

®

©

-

ª

®

9

F

9

:

max

ꢆ^F_>

×

(1 − ꢇ_>

)

Ö

F

©

-

ª

®

©

-

ª

®

F

9

max

ꢆ

×

(1 − ꢇ

)

>

9

9=1

9

«

¬

«

¬

9=1

«

¬

«

¬

subject to

ꢀ_*^ꢂ( 5 (-)) ≥ ꢆ_>, 9 = 1, 2, ..., :

ꢀ^ꢂ_!( 5₉(-)) ≥ ꢆ_>, 9 = :₁+ 1, :₁+ 2, ..., :

ꢁ_*^ꢂ( 5 (-)) ≤ ꢇ_>, 9 = 1, 2, ..., :

ꢁ_!^ꢂ( 5₉(-)) ≤ ꢇ_>, 9 = :₁+ 1, :₁+ 2, ..., :

0 ≤ ꢆ_>+ ꢇ_>≤ 1, 9 = 1, 2, ..., : , :₁+ 1, :₁+ 2, ..., :

0 ≤ ꢇ_>≤ ꢆ_>, 9 = 1, 2, ..., : , :₁+ 1, :₁+ 2, ..., :

3

(1−((* − 5 (-))/(* −! )))

3

9

1

4

+ ꢆ (1 − 4 ) ≥ 1, 9 = 1, 2, ..., :

9

>

9

1

9

(1−(( 5 (-)−! )/(* −! )))

3

9

+ ꢆ (1 − 4 ) ≥ 1, 9 = : + 1, : + 2, ..., :

>

1

9

(1−(( 5 (-)−(! −ꢄ ))/(ꢄ +ꢅ )))

3

9

(21)

+ ꢇ (1 − 4 ) ≤ 1, 9 = 1, 2, ..., :

>

1

9

1

9

(1−(((* +ꢄ )− 5 (-))/(ꢄ +ꢅ )))

3

9

+ ꢇ (1 − 4 ) ≤ 1, 9 = : + 1, : + 2, ..., :

>

1

9

0 ≤ ꢆ + ꢇ ≤ 1, 9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

1

9

0 ≤ ꢇ ≤ ꢆ

,

9 = 1, 2, ..., : , : + 1, : + 2, ..., :

>

1

9

1

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

9

8

- ≥ 0.

, (-) ≤ 2 + ꢃ(ℓ ), 8 = 1, 2, ..., <

1

8

- ≥ 0,

T heorem 3.2.1. A unique optimal solution (-^∗, ꢆ^∗, ꢇ^∗) of problem (22)

where F₉is the weight assigned to the 9th objective such that

F₉∈ (0, 1) and ₉₌₁F₉= 1.

Í

:

corresponds to a POS -^∗of problem (2), where ꢆ^∗= (ꢆ^∗_>, ꢆ^∗_>, ..., ꢆ^∗_>

)

1

2

:

1

Teferi H.T.,(2026)

48

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

and ꢇ^∗= (ꢇ^∗_>, ꢇ^∗_>, ..., ꢇ^∗_>).

Then, using (ii) and (iii), we get

1

2

:

1

Proof. Since

ꢆˆ_>≥ ꢆ^∗_>, ∀9 ∈ {1, 2, ..., : } and ꢆˆ_>> ꢆ^∗_>for at least one 9 ∈ {1, 2, ..., : }.

:

1

Ö

1

(iv)

/(ꢆ, ꢇ) =

ꢆ^F_>

×

(1 − ꢇ_>

)

9

©

-

ª

®

©

-

ª

®

9

F

9

9=1

ꢇ_>≤ ꢇ^∗_>⇒ 1 − ꢇ_>≥ 1 − ꢇ^∗_>, ∀9 ∈ {1, 2, ..., : },

«

¬

«

¬

ˆ

1

9

Let (-^∗, ꢆ^∗, ꢇ^∗) be a unique solution of problem (22). Then

(v)

ꢇ_>< ꢇ^∗_>⇒ 1 − ꢇ_>> 1 − ꢇ^∗_>, for at least one 9 ∈ {1, 2, ..., : }.

ˆ

1

9

/(ꢆ^∗, ꢇ^∗) > /(ꢆ, ꢇ), ∀(-, ꢆ, ꢇ) in the feasible space of (22).

Suppose -^∗is not a POS of problem (2). Then, by Deﬁnition 3.1.4, ∃ - in

Now, using (iv), we get

ˆ

the feasible space of (2) such that

:

1

Ö

ꢆˆ^F_>

>

(ꢆ^∗_>

)

9

F

9

.

(vi)

9

∗

ˆ

9=1

5 (-) ≥ 5 (- ), ∀9 = 1, 2, . . . , : and 5 (-) > 5 (- ), for at least one9 ∈ {1, 2, ..., : } (8)

1

9

Thus,

Using (v), we obtain

5₉(-^∗) − !₉

ˆ

5₉(-) − !₉

≥

,

∀9 = 1, 2, ..., : ,

1

*₉− !₉

:

1

Ö

and

F

(1 − ꢇ_>

)

>

(1 − ꢇ^∗_>

)

.

(vii)

9

ˆ

9

5₉(-) − !₉

5₉(-^∗) − !₉

ˆ

9=1

>

for at least one 9 ∈ {1, 2, ..., : }.

1

*₉− !₉

Finally, from (vi) and (vii), we conclude:

Since 3₉> 0,

∗

ˆ

9

:

4

^{−3 ((* − 5 (-))/(* −! ))}− 4⁻³

4

^{−3 ((* − 5 (- ))/(* −! ))}− 4⁻³

1

Ö

ꢆˆ^F_>

(1 − ꢇ_>

)

>

(ꢆ^∗_>

)

(1 − ꢇ^∗_>

)

=⇒

≥

,

∀9 ∈ {1, 2, ..., : },

9

F

9

F

1

9

ˆ

1 − 4⁻³

9

9=1

and

∗

ˆ

−

3

(

*

−

5

(

-

)

/

(

*

−

!

)

−

3

4

−

4

^{−3 ((* − 5 (- ))/(* −! ))}− 4⁻³

9

>

,

∗

ˆ

1 − 4⁻³

⇒ /(ꢆˆ, ꢇ) > /(ꢆ , ꢇ ),

9

for at least one 9 ∈ {1, 2, ..., : }.

Similarly, for non-membersh¹ip functions:

(ii)

which contradicts the optimality of (-^∗, ꢆ^∗, ꢇ^∗) for problem (22). Hence, no

∗

ˆ

such - exists, and therefore - is a POS of problem (2).

*₉− 5₉(-)

*₉− 5₉(-^∗)

ˆ

The theorem can be proved similarly for the remaining two perspectives of

≤

, ∀9 ∈ {1, 2, ..., : },

1

the DMs and for minimization objectives.

*₉− (!₉− ꢄ₉) *₉− (!₉− ꢄ₉)

and

The overall solution procedure of the proposed method for solving an

IFMOO problem can be summarized as follows:

ˆ

*₉− 5₉(-)

*₉− 5₉(-^∗)

<

,

for at least one 9 ∈ {1, 2, ..., : },

1

*₉− (!₉− ꢄ₉)

Step 1.

Step 2.

Formulate the IFMOO problem (1).

∗

ˆ

−

3

(

5

(

-

)

+

ꢄ

−

!

)

/

(

*

+

ꢄ

−

!

)

−

3

4

−

4

^{−3 (( 5 (- )+ꢄ −! )/(* +ꢄ −! ))}− 4⁻³

9

=⇒

≤

,

Transform the IFMOO problem into the equivalent crisp

MOOP (2) by employing the accuracy function.

1 − 4⁻³

9

∀9 ∈ {1, 2, ..., : } , and

4^{−3 (( 5 (-)+ꢄ −! )/(* +ꢄ −! ))}− 4⁻³

1

Step 3.

Step 4.

Step 5.

Solve each objective function independently and construct

the pay-oﬀ table to determine the lower and upper bounds,

denoted by !₉and *₉, respectively, for each objective

function 5₉(-), where 9 = 1, 2, ..., :.

∗

ˆ

^{−3 (( 5 (- )+ꢄ −! )/(* +ꢄ −! ))}− 4⁻³

9

4

<

,

1 − 4⁻³

9

for at least one 9 ∈ {1, 2, ..., : }.

(iii)

1

Deﬁne

Construct the exponential membership and non-membership

functions corresponding to each objective function under

the optimistic, pessimistic, and mixed approaches using

the tolerance parameters ꢄ₉and ꢅ₉, where 9 = 1, 2, ..., :.

!

ˆ

9

*

− 5 (-)

9

−3

9

*

−!

9

− 4⁻³

9

4

ꢆˆ_>

=

,

9

1 − 4⁻³

9

Relax or reduce the ≤ type and ≥ type constraints by

employing the assigned violation parameter ;₈in the forms

2₈+ ꢃ(;₈) and 2₈− ꢃ(;₈), respectively, where ꢃ ∈ (0, 1) and

8 = 1, 2, ..., <.

ꢀ

ꢁ

∗

*

− 5 (-

)

9

*

−!

− 4⁻³

9

ꢆ^∗_>

,

1 − 4⁻³

9

!

Step 6.

Step 7.

Develop and solve the problem according to the DM’s

preference under the optimistic (22) or pessimistic (23) or

mixed (24) model.

ˆ

9

5 (-)+ꢄ −!

9

*

+ꢄ −!

9

− 4⁻³

9

4

ˆ

ꢇ_>

=

,

9

1 − 4⁻³

9

If the obtained solution satisﬁes the DM, then terminate the

solution procedure. Otherwise, reformulate the problem

and repeat the process until a satisfactory solution is

obtained.

ꢀ

ꢁ

∗

5 (- )+ꢄ −!

9

*

+ꢄ −!

− 4⁻³

9

ꢇ^∗_>

.

1 − 4⁻³

9

Teferi H.T.,(2026)

49

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

By solving each objective independently, subject to the constraints, the

ideal solutions and the extreme values of the objectives are identiﬁed.

4 Comparative Analysis

To verify the eﬀectiveness of the proposed method, a comparative analysis

was conducted using the approach proposed by V. Singh and Yadav (2018).

The manufacturing problem presented in their study was employed for this

purpose. Accordingly, the transformed equivalent crisp problem is

Accordingly, - = (0, 84.09, 52.66) and - = (0, 120.85, 0) which give

1

!

= 1223.62, * = 1272.69, and ! = 468.3²0, * = 595.73.

1

2

For an optimisti¹c DM, problem (25) is transformed into a single-objective

problem of the form (26) using model (22), by assigning equal weights to

the objectives and considering violation parameters of 25 and 20 units for

the ﬁrst and second constraints, respectively.

max 5 (-) = 7.5G₁+ 10.125G₂+ 8G ,

1

min 5 (-) = 2.9375G₁+ 3.8750G₂+³5.1250G

2

3

subject to

2.9375G₁+ 2.0625G₂+ 2.9375G ≤ 328.125,

(25)

3

3.875G₁+ 2.9375G₂+ 2.0625G ≤ 355.625,

3

2.0625G₁+ 2.9375G₂+ 2.9375G ≥ 355,

3

- = (G , G , G ) ≥ 0.

1

2

3

max / = ꢆ⁰₁^.5ꢆ⁰₂^.5(1 − ꢇ )^0.5(1 − ꢇ )^0.5

1

2

subject to

+10.125G +8G

4^{0.2(1−((1272.69−(7.5G}

+3.8750G +5.1250G

3

^{))/(1272.69−1223.62)))}+ ꢆ (1 − 4^0.2) ≥ 1,

1

2

3

1

4^{0.2(1−((2.9375G}

^{−468.30)/(595.73−468.30)))}+ ꢆ (1 − 4^0.2) ≥ 1,

1

2

+10.125G +8G −(1223.62−(ꢃ(1272.69−1223.62))))/(1272.69−(1223.62−(ꢃ(1272.69−1223.62)))))

4^{0.2(1−((7.5G}

+

1

2

3

ꢇ (1 − 4^0.2) ≤ 1,

1

+3.8750G +5.1250G ))/((595.73+(ꢃ(595.73−468.30)))−468.30)))

4^{0.2(1−(((595.73+(ꢃ(595.73−468.30)))−(2.9375G}

+

1

2

3

ꢇ (1 − 4^0.2) ≤ 1,

2

2.0625G₁+ 3.8750G₂+ 2.9375G ≤ 333.125 + ꢃ(25),

3

3.8750G₁+ 2.0625G₂+ 2.0625G ≤ 365.625 + ꢃ(20),

(26)

3

2.9375G₁+ 2.0625G₂+ 2.9375G ≥ 360,

3

0 ≤ ꢆ₉+ ꢇ₉≤ 1, 9 = 1, 2

0 ≤ ꢆ₉≤ 1, 9 = 1, 2

0 ≤ ꢇ₉≤ 1, 9 = 1, 2

G , G , G ≥ 0.

1

2

3

Solving this problem, using LINGO (LINDO Systems Inc., 2017) version 21,

the solution is presented in Table 1.

Table 1: Solutions of problem (26) under diﬀerent ꢃ values.

ꢃ

-

5 (-)

, (-)

1

2

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(0.00, 121.49, 3.30)

(0.00, 123.79, 0.00)

(0.00, 124.46, 0.00)

(0.00, 125.14, 0.00)

(0.00, 125.70, 0.00)

1248.50

1253.35

1260.24

1267.13

1272.69

482.56

479.68

482.31

484.95

487.08

257.33

255.31

256.72

258.12

259.25

361.63

363.62

365.62

367.62

369.24

363.63

363.62

365.62

367.62

369.24

As ꢃ increases, 5 (-) improves while 5 (-) reaches its lowest value around

Similarly, by reformulating problem (25) for pessimistic and mixed DM

perspectives using models (23) and (24), respectively, we obtain the

following solutions presented in Tables 2 and 3, respectively.

1

2

ꢃ = 0.4 and then slightly worsens for larger values of ꢃ. This shows a

trade-oﬀ, where increasing emphasis shifts from minimizing 5 (-) toward

2

maximizing 5 (-). The solution becomes stable for larger ꢃ.

1

Table 2: Solutions of Problem (25) under the pessimistic perspective.

ꢃ

-

5 (-)

, (-)

1

2

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(0.00, 114.81, 11.82)

(0.00, 118.60, 7.39)

(0.00, 122.39, 2.96)

(2.02, 118.03, 6.35)

(2.91, 118.91, 4.40)

(3.47, 118.96, 4.23)

1256.98

1259.93

1262.88

1260.97

1260.91

1264.38

505.45

497.44

489.44

495.83

491.84

492.86

271.51

266.32

261.13

268.01

266.70

267.98

361.62

363.62

365.63

367.62

369.62

371.62

371.97

370.09

368.22

369.52

409.67

369.04

Teferi H.T.,(2026)

50

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

For the pessimistic variants of the problem, as ꢃ increases, 5 (-) improves

ﬂuctuations in both objectives.

while 5 (-) remains relatively stable in the mid-range of ꢃ, w¹ith only minor

2

Table 3: Solutions of Problem (25) under the mixed perspective.

ꢃ

-

5 (-)

, (-)

1

2

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(0.00, 114.81, 11.82)

(0.00, 118.60, 7.39)

(0.00, 119.26, 7.42)

(0.00, 125.15, 0.00)

(0.00, 125.49, 0.00)

(0.00, 125.56, 0.00)

1256.98

1259.93

1262.88

1267.13

1270.58

1271.13

505.45

497.44

500.15

484.95

486.27

486.54

271.51

266.32

267.76

258.12

258.82

258.96

361.62

363.62

365.62

367.62

368.62

368.82

371.97

370.09

372.12

367.62

368.62

368.82

For the mixed variants of the problem, as ꢃ increases, the solution steadily

ꢃ = 0.6, the solution minimizes 5 (-).

2

improves 5 (-), while 5 (-) stabilizes after an initial ﬂuctuation. Around

1

2

Table 4: Comparison of the proposed method with existing approach.

Variant

Proposed method (ꢃ = 0.3 to 0.8)

Existing method (from C = 1 to 5, under two reference

conditions)

Optimistic

Pessimistic

Mixed

5 (-) maximized within the range 1248.50 to 1272.69,

5 (-) maximized within the range 1244.91 to 1248.62,

1

5 (-) minimized within the range 487.08 to 479.68

5 (-) minimized within the range 530.80 to 520.33

2

5 (-) maximized within the range 1256.98 to 1264.38,

5 (-) maximized within the range 1248.63 to 1251.92,

1

5 (-) minimized within the range 505.45 to 489.44

5 (-) minimized within the range 539.37 to 530.7

2

5 (-) maximized within the range 1256.98 to 1271.13,

5 (-) = 1248.77 (maximized),

1

5 (-) minimized within the range 505.45 to 484.95

5 (-) = 530.66 (minimized)

2

The proposed method dominates the existing approaches across all

decision-making variants by achieving higher values of 5 (-) and lower

values of 5 (-) under diﬀerent values of ꢃ ∈ (0, 1). It sho¹ws a signiﬁcant

2

reduction in 5 (-), indicating improved minimization performance.

2

Generally, the results demonstrate that the proposed model provides more

eﬃcient and balanced trade-oﬀ between the two conﬂicting objectives.

5 Model Application

5.1 Description of the Study Area

The farming site is in Abeshge district, Gurage Zone, central Ethiopia,

between 8^◦19⁰–8^◦45⁰N latitude and 37^◦45⁰–38^◦7⁰E longitude (Nasir &

Hundie, 2014). Mean annual temperatures range from 18^◦C to 28.3^◦C, with

rainfall of 801–1400 mm, mostly during the Kiremt season (Dessie et al.,

2017). The soil is sandy loam, with a pH of 6.40–6.92.

Farming is mainly rainfed due to limited irrigation, though a few seasonal

rivers support perennial crops like mangoes and bananas along their

banks. Teﬀ (Eragrostis tef), maize (Zea mays), pepper (Capsicum annuum),

chickpea (Cicer arietinum), bean (Phaseolus vulgaris), and sorghum

(Sorghum bicolor) are the most widely cultivated crops in the area. These

crops dominate the farming pattern, accounting for about 85% of the

cropped area in the district.

The study area was chosen due to its high crop production potential and the

availability of accessible agricultural data. The considered LSF has detailed

information on existing cropping patterns, which makes the farming site

suitable for empirical analysis. Furthermore, the study area represents the

dominant farming system of the area, allowing the ﬁndings to be relevant

to smallholder farmers in the area. This LSF practices rainfed farming on

1,033ha of land in Gefersa kebele. The location map of the study area is

shown in Figure 4.

Figure 4: Map of the study area.

5.2 Methods of Data Collection and

Analysis

A stratiﬁed purposive sampling method was employed to intentionally

select the large-scale farm (LSF) site based on its suitability for the study.

The selection criteria included the availability of reliable and accessible

data, representative farming practices, adequate capital investment, and

suﬃcient manpower. These characteristics enabled the farm to eﬀectively

represent the target study population and provide meaningful insights into

the research objectives.

The existing farming pattern mainly targeted on the achievement of

maximum production. The required input allocation to the crops are mainly

determined by experience, even though they rarely apply the advice and

paradigm of the developmental agents and extension workers in the district.

Teferi H.T.,(2026)

51

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

The data were collected from farmers, managers, and workers of the

LSF, extension experts, development agents, the zonal rural development

oﬃce, meteorological stations, and marketing agencies through surveys,

questionnaires, and key informant interviews.

expenditures for cultivation and agricultural input resources are presented

in Tables 5 to 9.

Crop data are presented in Supplementary Materials S.7 and S.8. These,

along with the soil data in Supplementary Material S.9, are used to calculate

crop water requirements.

The dataset incorporates environmental, climatic, hydrological,

agricultural, and socio-economic factors. Hydrological data cover eﬀective

rainfall and crop water demand. Eﬀective rainfall was calculated using the

dependable rain (FAO/AGLW formula) method Allen et al., 1998, while

crop water demand was estimated with the Penman-Monteith equation

(Allen et al., 1998; Smith, 1992) using crop evapotranspiration.

The considered crops are denoted as 2₁for maize, 2₂for teﬀ,2₃for sorghum,

2

for pepper, 2 for chickpea, 2 for bean.

4

5

6

In Table 5, ꢃ

12

and ꢃ

represent the fertilizers required for crop 2₌, where

22

=

= = 1, 2, 3, 4, 5, 6, in the ﬁrst and second rounds, respectively. Accordingly,

ꢃ

denotes NPS, while ꢃ

represents Urea. Similarly, ꢄ and ꢄ

12 22

=

12

22

rep⁼resent the required herbicides for crop 2₌in the ﬁrst and second rounds,

respectively. The pesticide required for crop 2₌is denoted by %₂₌. Thus, the

following symbols are used to denote diﬀerent herbicides and pesticides:

=

Crop data such as rooting depth, crop coeﬃcient, critical depletion, yield

response, crop height, and crop calendar were obtained from FAO Manual

56 (Allen et al., 1998) and related literature.

Socio-economic data include crop market prices, labor and machinery costs,

fertilizer, herbicide, and pesticide dosages and prices, and land resource

information.

ꢄ

and ꢄ

for Atrazine, ꢄ

12

for 2,4-D, ꢄ

for Pallas 45 OD, ꢄ

22 12

22

12

1

3

2

4

for Glyphosate, ꢄ

%

and ꢄ

for Pendimethalin, ꢄ₂for S-metolachlor,

22

for Diazinon 60⁵EC, %₂for Dimethoate, %₂for Karate 5 EC, %₂for

6

2

Ethiozinon 60 EC, %₂for Highway 50 EC, and %³₂for Proﬁt.

1

2

4

5

6

The gathered data from diﬀerent sources for a speciﬁc parameter are

arranged into ﬁve groups, viz., the extreme minimum, minimum, median,

maximum, and extreme maximum values, based on the level of their

deviation from the median value. The values less than the median value

are arranged into the extreme minimum and minimum values. Speciﬁcally,

the average of the highly deviated values from the median value is taken

as the extreme minimum value, while the average of the relatively less

deviated values from the median is considered as the minimum value. In

the same manner, the values greater than the median value are arranged into

the maximum and extreme maximum values in the agricultural production

problem. The average of the median values of a parameter is taken as the

mean value.

Table 5: Usage of fertilizer, herbicide and pesticide.

ꢃ

ꢄ

%

2

(!/ℎ0)

12

22

12

22

=

Crop

(:,/ℎ0)

(!/ℎ0)

g

e

maize (2 )

100

125

-

1

g

e

f

e

teﬀ (2 )

100

1

0.5

1

2

g

e

sorghum (2 )

100

-

1

3

g

f

pepper (2 )

100

200

1.5

-

1.5

4

g

e

chickpea (2 )

100

-

1

5

g

e

g

e

bean (2 )

100

1

0.75

1

6

The IFN in Table 5 describe the following numbers.

Using this principle, the aggregated values are used to construct triangular

IFNs to represent the APP problem more realistically.

g

f

200 = h196, 198, 208; 194, 198, 210i,

1.5 = h1.1, 1.6, 1.7; 1, 1.6, 1.8i,

g

e

125 = h123, 125, 127; 122, 125, 128i,

1 = h0.75, 1, 1.25; 0.5, 1, 1.5i,

g

100 = h97, 99, 105; 95.5, 99, 106.5i,

0.75 = h0.6, 0.7, 1; 0.3, 0.7, 1.3i,

In this study, the last twelve years climatic data from 2013 to 2024 were

gathered from Emdbir meteorological station, with an altitude of 2082<,

latitude 8.13^◦N and longitude 37.93^◦E.

In order to handle the current extreme variation in the cost of input

resources for crop cultivation and crop prices in the country, we have

employed closely related data for the proposed study. Accordingly, the

average data from 2022 to 2024 of the existing cropping schemes, crop yield

(FAO, 2022, 2023), labor force, crop prices, cost of seeds, and other related

f

e

2.5 = h2, 2.6, 2.8; 1.9, 2.8, 2.9i,

2 = h1.9, 2, 2.1; 1.5, 2, 2.5i

f

0.5 = h0.2, 0.5, 0.7; 0.1, 0.5, 1i.

The costs of cultivation, including plowing, threshing, land rent, seed, labor,

herbicides and pesticides, and fertilizers, are provided in Supplementary

Materials S.2, S.3, S.4, S.11, and S.12, respectively.

The overall cost of the cultivation of each crop is presented in Table 6.

Table 6: Cost of cultivation (ꢁ₂) and crop yield (. ₌).

2

=

ꢁ₂

.

(@C;/ℎ0)

2

=

Crop

(ꢂ)ꢅ/ℎ0)

maize (2 )

h35623, 35627, 35628; 35620, 35627, 35629i

h35698, 35705, 35714; 35689, 35705, 35719i

h26838, 26845, 26854; 26829, 26845, 26859i

h41113, 41120, 41129; 41104, 41120, 41134i

h20655, 20659, 20666; 20649, 20659, 20674i

h25124, 25131, 25140; 25115, 25131, 25145i

h50, 54, 62; 47, 54, 65i

1

teﬀ (2 )

h12.5, 13.5, 16.5; 12, 13.5, 17i

h23, 24.5, 27.5; 22.5, 24.5, 29i

h11, 13.5, 14; 10, 13.5, 16i

h16.5, 17.5, 20; 15.5, 17.5, 22i

h21, 23.5, 24; 20, 23.5, 25i

sorgh²um (2 )

pepper (2 )³

4

chickpea (2 )

5

bean (2 )

6

The climatic data are used to calculate the crop water requirements, which are presented in Supplementary Material S.10.

Table 7: Seed (ꢂ₂), labor (ꢅ₂) and water requirement (,₂₌).

=

ꢂ₂

(:,/ℎ0)

ꢅ₂

(<3/ℎ0)

,

2

(< /ℎ0)

=

Crop

3

maize (2 )

h22, 26, 25; 17, 26, 27i

h29, 30, 30.5; 28.5, 30, 32i

h14, 14.5, 16.5; 13.5, 14.5, 18i

h15, 15.5, 17.5; 14.5, 15.5, 19i

h26, 28, 30; 25, 28, 31i

h19, 21, 23; 18, 21, 24i

h80, 86, 89; 75, 86, 92i

h70, 76, 80; 64, 76, 82i

h35, 41, 45; 30, 41, 46i

h97, 99, 105; 94, 99, 108i

h25, 31, 35; 20, 31, 36i

h47, 51, 52; 45, 51, 55i

h3570, 3610, 3625; 3565, 3610, 3648i

h2580, 2625, 2640; 2564, 2625, 2660i

h3240, 3249, 3252; 3225, 3249, 3255i

h4750, 4766, 4770; 4705, 4766, 4783i

h2515, 2540, 2560; 2505, 2540, 2572i

h2355, 2360, 2386; 2347, 2360, 2400i

1

teﬀ (2 )

sorgh²um (2 )

pepper (2 )³

4

chickpea (2 )

5

bean (2 )

6

Teferi H.T.,(2026)

52

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

The farming machines used in the area are a tractor and a combine harvester. Machine hours for plowing are considered until the land is ready for sowing;

details are in Supplementary Material S.1. The required total machine hours for each crop are shown in Table 8.

Table 8: Machine hours requirement ("₂) and proﬁt (#₂₌).

=

"

(ℎA/ℎ0)

#

2

(ꢂ)ꢅ/ℎ0)

2

=

Crop

maize (2 )

h1.83, 2.167, 2.33; 1.42, 2.167, 2.42i

h1.75, 2, 2.42; 1.5, 2, 2.5i

h33117, 33123, 33136; 33109, 33123, 33138i

h13293, 13295, 13297; 13290, 13295, 13300i

h18148, 18154, 18165; 18141, 18154, 18170i

h192873, 192881, 192888; 192862, 192881, 192893i

h25830, 25842, 25849; 25820, 25842, 25853i

h16262, 16269, 16278; 16253, 16269, 16283i

1

teﬀ (2 )

sorgh²um (2 )

pepper (2 )³

h1.75, 2, 2.42; 1.5, 2, 2.5i

4

chickpea (2 )

h1.83, 2.167, 2.33; 1.42, 2.167, 2.42i

h1.75, 2, 2.42; 1.5, 2, 2.5i

5

bean (2 )

6

The cropland allocation in the existing situation during 2020 - 2024 is given in the Supplementary Material S.5. Accordingly, the average land allocation of

the existing system and the attainability and non-attainability degrees of the intended objectives are depicted in Table 9. Where G , G , G , G , G , and G

6

2

3

5

denote the land area allocated to maize, teﬀ, sorghum, pepper, chickpea, and beans, respectively. The functions 5 (-), 5 (-), 5 (-),¹and 5 (-),⁴respectively,

1

2

3

4

represent the yield, proﬁt, cost, and labor objectives.

Table 9: Existing farm pattern.

- = (G , G , G , G , G , G )

5 (-)

6⁴4160

0.20

4

1

2

3

(221.02, 141.41,¹401².32,³106.42,⁵62.⁶29, 121.57)

29470.15

41510180

31413470

ꢀ

ꢁ

0.81

0.19

0.20

0.28

0.17

0.30

0.28

Some mathematical software is employed to solve the considered APP

problem and assist in analyzing the results of the study. CROPWAT

8.0 software (Smith, 1992) is used to generate and analyze the water

requirement of crops, and LINGO (LINDO Systems Inc., 2017) is used to

solve complex mathematical problems.

are described as IFN. Thus, the objectives and constraints of the intended

problem are expressed as intuitionistic fuzzy functions. Accordingly,

IFMOO model is employed for comprehensive and eﬃcacious investigation.

The Kiremt season is the widely practiced cropping season in the study area.

Crops such as maize, pepper, and sorghum are sown during the Belg season

and harvested by the end of Kiremt. Teﬀ and beans are sown in the Kiremt

season and harvested during the Bega season, while chickpea is planted in

the last days of Kiremt and harvested in the Bega season. Thus, all six crops

considered in this study are generally cultivated within a single cropping

season.

5.3 Problem Formulation

As a result of the inconsistent nature and imprecision of the pertained

agricultural data, crop planning rests under the inﬂuence of risk and

uncertainty (Luo et al., 2023). So the parameters in the APP problem

Table 10: Decision variable and parameters.

Parameters

)!

)!_A

Description

Total farmland for crop cultivation (ℎ0)

Minimum land area required for cultivation (ℎ0)

Average yield per unit area of crop 2 (@C;/ℎ0)

.

2

#

Average net proﬁt of crop 2 per hectare (ꢂ)ꢅ/ℎ0)

Average investment per unit area of crop 2 (ꢂ)ꢅ/ℎ0)

Average machine-hours required per unit area of crop 2 (ℎA/ℎ0)

Average labor required per unit area of crop 2 (<3/ℎ0)

Average water requirement per unit area of crop 2 (<³/ℎ0)

2

ꢁ₂

"

2

ꢅ₂

,

2

ꢂ₂

ꢃ_;2

%

;2

ꢄ_;2

)"

),

)ꢂ₂

)ꢃ_;

Seed required per unit area of crop 2 (:,/ℎ0)

Cℎ

;

type of fertilizer required per unit area for crop 2 (:,/ℎ0)

Cℎ

type of pesticide required per unit area for crop 2 (!/ℎ0)

Cℎ

;

type of herbicide required per unit area for crop 2 (!/ℎ0)

Total available machine-hours (ℎA)

Total available water (<³)

Total available seed for crop 2 (:,)

Total available ;^Cℎfertilizer (:,)

)%_;

)ꢄ_;

Total available ;^Cℎpesticide (!)

Total available ;^Cℎherbicide (!)

Decision

Description

Land area allocated to crop 2 (ℎ0)

variable

-

2

Note: <3 denotes man-days.

Teferi H.T.,(2026)

53

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Objective functions

during growth stage from diﬀerent pests and insects. This constraint is

formulated as

Based on their accessibility and regional importance, the following four

objectives are considered in this study.

6

Õ

f

g

%

_;2-₂- )%_;, ; = 1, 2, ..., !.

(34)

(i) Crop yield achievement

2=1

The estimated yield of a crop is equal to the product of the cultivable area

of land and the average yield produced per unit area of land.

Thus, the maximization of the total yield of the considered six crops can be

expressed as

(vi) Machine hours

Diﬀerent types of machines are needed for various tasks of agriculture,

such as tilling, plowing, sowing, cultivating, harvesting, threshing, etc. The

sum of the machine hours allocated to each season should not exceed the

machine hours required in a year. That means,

6

Õ

e

max 5 (-) ≈

. -₂.

(27)

1

2

2=1

6

Õ

(ii) Net proﬁt goal

g

"₂-₂- )".

(35)

The net proﬁt of various crops, is the product of the net proﬁt of each crop

per unit area of land and its respective utilized land, which is described as

2=1

(vii) Cultivable land availability

6

Õ

e

f

The sum of cultivable land for all crops must not exceed the total available

land. Furthermore, the total cultivable land should be less than the entire

arable land available in the study area. This is formulated as

max 5 (-) ≈

#₂-₂.

(28)

2

2=1

(iii) Cost of cultivation goal

6

To get the optimum production, farmers should invest a certain amount

of money for land rent, fertilizers, seeds, herbicides, pesticides, rental

machines, labor force, etc. Minimizing this working capital is another

important objective of farmers and mathematically given by

Õ

f

-₂- )!.

(36)

2=1

On the other hand, a minimum cultivable land area should be allocated to

crop production to maintain a minimum level of agricultural output and

proﬁt while ensuring eﬃcient utilization of available land resources. This

constraint is mathematically expressed as

6

Õ

e

min 5 (-) ≈

ꢁ₂-₂.

(29)

3

2=1

(vi) Labor requirement

The labor objective is described as

6

Õ

g

-₂¥ )!_A.

(37)

2=1

=

Õ

e

f

min 5 (-) ≈

ꢅ₂-₂.

(30)

4

(i) Seed requirement

2=1

Seed availability constraints were incorporated into the model to describe

the limitations in accessing selected seed for each crop. The maximum seed

availability for each crop was determined based on the recommended seed

requirement per hectare (Table 7), the maximum feasible cultivable area

for each crop, and crop suitability conditions under local agro-ecological

settings (Semu et al., 2022). This constraint mathematically expressed as

Constraints

The above four objectives are subject to the following eight constraints.

(ii) Water requirement

Additional water through irrigation is required to meet the crop’s

evapotranspiration needs and optimize yield. The constraint for water

supply can be described as

f

g

ꢂ₂-₂- )ꢂ₂, 2 = 1, 2, ..., 6.

(38)

6

Õ

f

g

,₂-₂- ),.

(31)

(viii) Non-negativity

2=1

In the modelling APP problem, all decision variables should be

non-negative.

-

≥ 0

2 = 1, 2, ..., 6.

(39)

2

For sustainable optimal crop yield and maximum proﬁt, agricultural input

resources should not be used at the expense of the environment (Li et al.,

2020). Therefore, to reduce the adverse eﬀects of fertilizers, pesticides, and

herbicides on the environment, the optimum amounts of these inputs must

be considered alongside the utilization of other favorable resources.

(iii) Dosage of fertilizer

The stated objectives and constraints align with economic, environmental,

and social goals, with aims to boost net beneﬁts, use resources wisely to

limit environmental harm, and increase local jobs (Li et al., 2020; Zhang &

Georgescu, 2022).

To maintain and improve the productivity of the soil, diﬀerent types of

fertilizers have to be used optimally according to the characteristics of the

crops, soil type and climate of the region. This constraint is expressed as

In this study, the three variants of the problem are considered

independently to address the interference of uncontrolled conditions and

associated risks, and individual diﬀerences in the DM process. This assists

farmers and managers from diﬀerent perspectives by proposing various

possible alternative management schemes.

6

Õ

f

g

ꢃ_;2-₂- )ꢃ_;, ; = 1, 2.

(32)

2=1

In the optimistic assumption, a farmer considers using farming resources

with varying degrees of acceptance and ﬂexibility to accommodate other

possible alternative operations, presuming that it oﬀers certain beneﬁts.

Conversely, a pessimist DM tends to be skeptical about implementing all

possible alternative farming tasks and partially considers those with a lower

degree of acceptance. From a mixed perspective, the farmer’s assumption

lies between the optimistic and pessimistic viewpoints.

As a result, the solution of the APP problem varies according to each

perspective (Kis et al., 2021), which leads to diﬀerences in the determination

of the OCP. Each viewpoint has its advantages and disadvantages.

Therefore, this study considers the optimistic, pessimistic, and mixed

aspects of DM for eﬃcient management of agricultural resources.

(iv) Amount of Herbicide

DM requires a certain level of herbicides for several crops during growth

stage for better yield taking the availability and environmental aspects into

account. Mathematically described as

6

Õ

g

ꢄ_;-₂- )ꢄ_;, ; = 1, 2, ..., !.

(33)

2

2=1

(v) Dosage of pesticide

Pesticide is another essential input resource of agriculture to protect crops

Teferi H.T.,(2026)

54

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

In this study, from 1, 033ℎ0 land of the LSF, nearly 1, 017ℎ0 of farmland

is considered for crop cultivation, and 6ℎ0 to 7ℎ0 of land is supposed to

be left permanently for forestation to maintain the ecological balance of

the environment. In the existing system, 5ℎ0 of cropland is occupied by

perennial crops. About 1.50ℎ0 of land of the LSF is permanently left over

as a residential place for workers and roads. Moreover, 2.5ℎ0 of land is not

suitable for farming and is currently used for animal grazing.

The land management of the LSF is a traditional approach based on the

weather conditions. They interchangeably use plots of land for diﬀerent

crops, and there is no reasonable pattern of operation on cropland.

In this work, three scenarios of crop cultivation are designed based on

the conventional cultivation pattern and suitability of the devised farming

system in the study area.

third scenarios are relaxed based on resource availability.

In the problem formulation of the three scenarios, the upper and lower

limits of the constraints, the value of violations, and tolerances are mainly

based on the availability of agricultural resources. For the second and third

scenarios, the land constraint attributed to chickpeas is expressed in terms

of the remaining crops.

The weights assigned to the objectives and constraints are estimated

based on the preferences rated by managers of the LSF, farmers, and

developmental agents of the study area.

6 Results and Discussions

We denote the land areas allocated to crop 2 by G₂, where 2 = 1, 2, ..., 6,

representing the land areas of maize, teﬀ, sorghum, pepper, chickpea, and

bean, respectively.

The ﬁrst scenario considers the case when chickpeas are planted on

the land leftover in the ﬁrst season of crop cultivation. Thus, in this

scenario, chickpea faces the same land rent as other crops. This system is

mainly adopted to fertilize unplowed land and to make use of ﬁelds left

uncultivated in the ﬁrst season due to factors like irregular rainfall, labor

shortages, limited seed varieties, and lack of capital or fertilizers.

The functions 5₉(-), where 9 = 1, 2, 3, 4, denote the objectives of production,

proﬁt, expenditure, and manpower, respectively. The constraints are

denoted by ,₈(-), where 8

=

1, 2, ..., 8, corresponding respectively to

water, ﬁrst-round fertilizer, second-round fertilizer, ﬁrst-round herbicide,

In the second scenario, farmers plant chickpeas on land right after

harvesting maize. This common practice helps to use the fertile soil left by

maize and saves farmers from the additional cost of renting extra land for

smallholder farmers with limited ﬁelds to cultivate multiple crops. Based

on the existing farming system in the study area, at least 33% of the land

allocated to maize is subsequently used for chickpea cultivation after maize

harvest.

second-round herbicide, pesticide, machine hours, and the maximum

available land area.

In addition, minimum land area cultivation

requirements are imposed to secure the minimum proﬁt and yield, while

seed constraints are included to account for limitations in seed availability.

Based on the available agricultural resources, the violation parameters of

the constraints are assigned as ℓ = 5235, ℓ = 935, ℓ = 940, ℓ = 96,

1

4

ℓ

= 88, ℓ = 78, ℓ = 104, and ℓ = 4. The vio²lation para³meters for the seed

In the third scenario, chickpeas are planted after harvesting maize,

sorghum, and beans. Like the second scenario, the most important reason

for farmers to use this cropping plan is to increase the yield from fertile land

and minimize additional cultivation expenses. In order to eﬀectively utilize

the land for chickpea after harvesting maize, sorghum, and bean, the land

allocated to chickpea should not be less than the combined land allocated to

teﬀ and pepper. Furthermore, the land allocated to bean should be at least

equal to that allocated to sorghum. However, according to the farmers, this

scenario is rarely practiced as land preparation for the succeeding crop after

harvesting sorghum is a relatively challenging activity.

c⁵onstrain⁶ts of mai⁷ze, teﬀ, sorghum, pepper, chickpea, and bean are assigned

as 66, 40, 10, 10, 44, and 38, respectively, based on their availabilities.

8

The weights assigned to the production, proﬁt, cost, and labor force

objectives are 30%, 28%, 28%, and 14%, respectively. Utilizing all these

values, the formulated problem is solved under optimistic, pessimistic, and

mixed viewpoints by employing models (22), (23), and (24), respectively.

The problem is deﬁned using tolerance values expressed as multiples of ꢃ,

with violations evaluated under the three approaches. To explore diﬀerent

solutions, several values of ꢃ ∈ (0, 1), speciﬁcally 0.30, 0.40, 0.50, 0.60, 0.70,

and 0.80, are considered in the solution process. Accordingly, the APP

problem is solved for the three scenarios based on the objectives and

constraints outlined in Section 5.3 and the data presented in Tables 5 to

8.

As chickpea is cultivated as a second crop after harvesting other crops

within the same cropping season, the land rent for chickpea under the

second and third scenarios decreases signiﬁcantly by 65% relative to the

annual land lease cost. Thus, to eﬀectively utilize the available land in this

production cycle, the upper limits of some constraints in the second and

Scenario 1: When chickpea is planted on the fallow land.

Teferi H.T.,(2026)

55

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

The mathematical expression of this problem has the following form:

e

f

g

f

g

f

g

max 5 (-) = 56G₁⊕ 14.5G₂⊕ 25G₃⊕ 13.5G₄⊕ 18G₅⊕ 23.5G ,

1

6

e

max 5 (-) = 33125G₁⊕ 13295G₂⊕ 18154G₃⊕ 192881G₄⊕ 24342G₅⊕ 16269G ,

2

6

e

min 5 (-) = 35625G₁⊕ 3705G₂⊕ 26845G₃⊕ 41119G₄⊕ 20659G₅⊕ 25132G ,

3

6

e

f

g

f

min 5 (-) = 84G₁⊕ 74G₂⊕ 41G₃⊕ 100G₄⊕ 31G₅⊕ 50G

4

6

subject to

3610G₁⊕ 2620G₂⊕ 3245G₃⊕ 4765G₄⊕ 2540G₅⊕ 2365G - 3410610,

6

g

100G₁⊕ 100G₂⊕ 100G₃⊕ 100G₄⊕ 100G₅⊕ 100G - 101700,

6

g

125G₁⊕ 100G₂⊕ 100G₃⊕ 200G - 108055,

4

e

f

e

(40)

1G₂⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1145,

6

e

f

e

g

1G₁⊕ 0.5G₂⊕ 1G₃⊕ 0.75G - 1017,

6

e

f

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1100,

6

g

e

g

e

2.67G₁⊕ 2.67G₂⊕ 2G₃⊕ 2G₄⊕ 2.67G₅⊕ 2G - 2205,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G₅⊕ 1G - 1017,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G₅⊕ 1G ¥ 1015,

6

f

g

26G - 21611, 30G - 15255, 14.5G - 99156,

1

2

3

g

f

15.5G - 6509, 28G - 15662, 20G - 12204,

4

5

6

- = (G , G , G , G , G , G ) ≥ 0,

1

2

3

4

5

6

where,

1145 = h1140, 1145, 1146; 1139, 1145, 1147i,

1017 = h1016, 1017, 1018; 1015, 1017, 1019i,

1015 = h1014, 1015, 1016; 1013, 1015, 1017i,

1100 = h1099, 1101, 1107; 1098, 1101, 1108i,

e

2205 = h2200, 2205, 2207; 2197, 2205, 2208i,

1 = h0.95, 0.99, 1; 0.94, 0.99, 1.15i,

3410610 = h3410609, 3410610, 3410615; 3410607, 3410610, 3410617i,

101700 = h101697, 101699, 101705; 101695, 101699, 101707i,

108055 = h108052, 108057, 108058; 108050, 108057, 108060i,

21611 = h21605, 21610, 21611; 21604, 21610, 21612i,

15255 = h15250, 15255, 15256; 15249, 15255, 15257i,

99156 = h99151, 99156, 99157; 99150, 99156, 99158i,

15662 = h15657, 15662, 15663; 15656, 15662, 15664i,

12204 = h12199, 12204, 12205; 12198, 12204, 12206i,

28138 = h28132, 28138, 28140; 28130, 28138, 28142i,

6509 = h6504, 6509, 6510; 6503, 6509, 6511i.

The remaining values of the coeﬃcients of the variables are presented in

Tables 5 to 8.

The solutions of problem (40) obtained using the proposed optimistic

approach for diﬀerent ꢃ ∈ (0, 1) are presented in Table 11.

Table 11: Solutions of problem (40) under an optimistic perspective.

ꢃ

-

5 (-)

5⁴4150

54630

55064

55464

55761

55751

/

1

2

3

0.3

0.4

0.5

0.6

0.7

0.8

(153.44, 0, 314.97, 78.79, 152.82, 314.97)

(162.93, 0, 305.66, 81.88, 158.86, 305.66)

(171.96, 0, 296.48, 84.73, 165.36, 296.48)

(180.58, 0, 287.40, 87.37, 172.26, 287.40)

(187.95, 0, 278.65, 89.34, 180.41, 278.65)

(191.60, 0, 270.90, 89.31, 192.28, 270.90)

27333.18

27557.00

27766.47

27963.23

28121.20

28163.46

34842180

35579290

36269800

36919230

37441020

37578060

28234760

28340800

28436290

28522470

28579940

28551380

0.2651

0.2811

0.2949

0.3070

0.3176

0.3274

From the above solutions, Table 11, the better compromised solution is obtained when ꢃ = 0.8. So, the compromised solution for the ﬁrst scenario is

- = (191.60, 0, 270.90, 89.31, 192.28, 270.90) and its detail is depicted in Table 12.

Teferi H.T.,(2026)

56

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Table 12: Compromise solution of problem (40).

- = (G , G , G , G , G , G )

5 (-)

5⁴5751

0.31

2

5

6

1

2

3

(191.60, 0, 27¹0.90, 89³.31,⁴192.28, 270.90)

28163.46

37578060

28551380

ꢀ

ꢁ

0.48

0.20

0.50

0.20

0.37

0.26

0.29

- = (G , G , G , G , G , G )

, (-)

2

5

6

2

4

(191.60, 0, 27¹0.90, 89³.31,⁴192.28, 270.90)

31¹24448

101500

6³8902

597.15

- = (G , G , G , G , G , G )

, (-)

2

5

6

5

7

8

(191.60, 0, 27¹0.90, 89³.31,⁴192.28, 270.90)

665.68

10⁶59.65

2287

1015

Based on the results given in Table 12, the required input resources to cultivate the crops under the allocated cropland are presented in Table 13.

Table 13: Input resource requirements for allocated crops under scenario one.

Crop

Seed (kg)

4789.95

4063.58

1428.93

5383.90

5418.11

ꢃ

12

(kg)

ꢃ

(kg)

ꢄ

(L)

ꢄ

153.44

314.97

(L)

%

2

=

(L)

22

12

22

=

maize (2 )

15344

31497

7879

23949.79

-

153.44

314.97

133.96

152.82

314.97

1

sorghum (2 )

27090.56

pepper (2 )³

17861.63

133.96

192.28

270.90

-

4

chickpea (2 )

15282

31497

-

5

bean (2 )

203.17

6

Scenario 2: Chickpea is planted after harvesting maize.

The mathematical expression of this problem has the following form:

e

f

g

f

g

f

g

max 5 (-) = 56G₁⊕ 14.5G₂⊕ 25G₃⊕ 13.5G₄⊕ 18G₅⊕ 23.5G ,

1

6

e

max 5 (-) = 36100G₁⊕ 13295G₂⊕ 18154G₃⊕ 192881G₄⊕ 29865G₅⊕ 16269G ,

2

6

e

min 5 (-) = 32650G₁⊕ 3705G₂⊕ 26845G₃⊕ 41119G₄⊕ 15135G₅⊕ 25132G ,

3

6

e

f

g

f

min 5 (-) = 84G₁⊕ 74G₂⊕ 41G₃⊕ 100G₄⊕ 31G₅⊕ 50G

4

6

subject to

3610G₁⊕ 2620G₂⊕ 3245G₃⊕ 4765G₄⊕ 2540G₅⊕ 2365G - 3667300,

6

g

100G₁⊕ 100G₂⊕ 100G₃⊕ 100G₄⊕ 100G₅⊕ 100G - 105900,

6

g

125G₁⊕ 100G₂⊕ 100G₃⊕ 200G - 127125,

4

e

f

e

1G₂⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1145,

6

(41)

e

f

e

g

1G₁⊕ 0.5G₂⊕ 1G₃⊕ 0.75G - 1017,

6

e

f

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1525,

6

e

2.167G₁⊕ 2.167G₂⊕ 2G₃⊕ 2G₄⊕ 2.167G₅⊕ 2G - 2715,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G - 1017,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G ¥ 1015,

6

e

g

1G ¥ 1G , 1G ¥ 0.33G ,

1

5

1

f

g

26G - 21611, 30G - 15255, 14.5G - 99156,

1

2

3

g

f

15.5G - 6509, 28G - 15662, 20G - 12204,

4

5

6

- = (G , G , G , G , G , G ) ≥ 0,

1

2

3

4

5

6

where

3667300 = h3667293, 3667304, 3667307; 3667290, 3667304, 3667310i,

g

2715 = h2712, 2715, 2718; 2710, 2715, 2720i, 0.33 = h0.31, 0.33, 0.35; 0.30, 0.33, 0.36i,

105900 = h105895, 105900, 105905; 105890, 105900, 105910i,

127125 = h127122, 127125, 127128; 127120, 127125, 127130i,

29490 = h29491, 29492, 29498; 29488, 29492, 29499i,

36100 = h36096, 36098, 36104; 36095, 36098, 36105i,

15135 = h15132, 15134, 15140; 15130, 15134, 15142i,

29865 = h29862, 29865, 29868; 29860, 29865, 29870i,

32650 = h32648, 32650, 32655; 32645, 32650, 32660i,

1525 = h1522, 1525, 1528; 1520, 1525, 1530i.

Teferi H.T.,(2026)

57

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

By solving problem (41) using the proposed method for the optimistic

variants of the problem, a better compromise solution is obtained for ꢃ =

0.8. The resulting compromised optimal solution for the second scenario

is - = (156.00, 0.00, 661.58, 122.85, 51.48, 74.57), with details presented in

Table 14.

Table 14: Compromise solution of problem (41).

- = (G , G , G , G , G , G )

5 (-)

5⁴7384

0.54

1

4

6

1

2

3

(156.00, 0.00, 661.²58, ³122.85,⁵51.48, 74.57)

29358.32

44088180

30558420

ꢀ

ꢁ

0.92

0.03

0.30

0.36

0.26

0.18

- = (G , G , G , G , G , G )

, (-)

1

4

6

2

3

4

(156.00, 0.00, 661.²58, ³122.85,⁵51.48, 74.57)

36¹01808

106648

110228

310.32

- = (G , G , G , G , G , G )

, (-)

1

4

6

5

8

(156.00, 0.00, 661.²58, ³122.85,⁵51.48, 74.57)

873.51

11⁶27.90

22⁷71.97

1015

The required input resources to cultivate the crops under the allocated cropland are presented in Table 15.

Table 15: Input resource requirements for allocated crops under scenario two.

Crop

Seed (kg)

3900.00

9923.75

1965.58

1441.44

1491.36

ꢃ

(kg)

ꢃ

(kg)

ꢄ

(L)

ꢄ

156.00

661.58

(L)

%

2

=

(L)

12

22

19500

12

22

=

maize (2 )

15600

66158

12285

5148

-

156.00

661.58

184.27

51.48

1

sorghum (2 )

66158

24570

pepper (2 )³

184.27

51.48

74.57

-

4

chickpea (2 )

-

5

bean (2 )

7457

55.93

74.57

6

Scenario 3: Chickpea is planted after harvesting maize, sorghum and beans.

The mathematical expression of this scenario has the following form:

e

f

g

f

g

f

g

max 5 (-) = 56G₁⊕ 14.5G₂⊕ 25G₃⊕ 13.5G₄⊕ 18G₅⊕ 23.5G ,

1

6

e

max 5 (-) = 36100G₁⊕ 13295G₂⊕ 21130G₃⊕ 192881G₄⊕ 29865G₅⊕ 19245G ,

2

6

e

min 5 (-) = 32650G₁⊕ 3705G₂⊕ 23870G₃⊕ 41119G₄⊕ 15135G₅⊕ 22155G ,

3

6

e

f

g

f

min 5 (-) = 84G₁⊕ 74G₂⊕ 41G₃⊕ 100G₄⊕ 31G₅⊕ 50G

4

6

subject to

3610G₁⊕ 2620G₂⊕ 3245G₃⊕ 4765G₄⊕ 2540G₅⊕ 2365G - 3667300,

6

g

100G₁⊕ 100G₂⊕ 100G₃⊕ 100G₄⊕ 100G₅⊕ 100G - 105900,

6

g

125G₁⊕ 100G₂⊕ 100G₃⊕ 200G - 127125,

4

e

f

e

1G₂⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1145,

6

(42)

e

f

e

g

1G₁⊕ 0.5G₂⊕ 1G₃⊕ 0.75G - 1017,

6

e

f

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1.5G₄⊕ 1G₅⊕ 1G - 1525,

6

e

2.167G₁⊕ 2.167G₂⊕ 2G₃⊕ 2G₄⊕ 2.167G₅⊕ 2G - 2715,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G - 1017,

6

e

1G₁⊕ 1G₂⊕ 1G₃⊕ 1G₄⊕ 1G ¥ 1015,

6

e

1G - 1G₁⊕ 1G₃⊕ 1G , 1G ¥ 1G₂⊕ 1G , 1G ¥ 1G .

5

6

5

4

6

3

f

g

26G - 21611, 30G - 15255, 14.5G - 99156,

1

2

3

g

f

15.5G - 6509, 28G - 15662, 20G - 12204,

4

5

6

- = (G , G , G , G , G , G ) ≥ 0,

1

2

3

4

5

6

where 21130 = h21126, 21130, 21134; 21124, 21130, 21136i,

23870 = h23864, 23870, 23876; 23862, 23870, 23878i,

19245 = h19240, 19245, 19247; 19237, 19245, 19248i,

22155 = h22153, 22155, 22160; 22152, 22155, 22163i.

Teferi H.T.,(2026)

58

East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

By solving problem (42) using the proposed method for the optimistic

- = (409.19, 0.00, 277.16, 51.48, 51.48, 277.16), with details presented in

variants of the problem, a compromise solution is obtained for ꢃ

=

Table 16.

0.8. The resulting compromised optimal solution for the third scenario is

Table 16: Compromise solution of problem (42).

- = (G , G , G , G , G , G )

5 (-)

6⁴6522

0.38

1

4

6

1

2

3

(409.19, 0.00, 277.²16, ³51.48, 5⁵1.48, 277.16)

37405.40

37428600

29013310

ꢀ

ꢁ

0.41

0.24

0.72

0.10

0.33

0.28

, (-)

8³9162

, (-)

24⁷41.61

0.26

- = (G , G , G , G , G , G )

, (-)

1

4

6

2

4

(409.19, 0.00, 277.²16, ³51.48, 5⁵1.48, 277.16)

34¹06694

106648

405.86

- = (G , G , G , G , G , G )

, (-)

1

4

6

5

8

(409.19, 0.00, 277.²16, ³51.48, 5⁵1.48, 277.16)

894.23

10⁶92.22

1015

The required input resources to cultivate the crops under the allocated cropland are presented in Table 17.

Table 17: Input resource requirements for allocated crops under scenario three.

Crop

Seed (kg)

10229.87

4157.44

823.68

1441.44

5543.25

ꢃ

(kg)

ꢃ

(kg)

ꢄ

(L)

ꢄ

409.19

277.16

(L)

%

2

=

(L)

12

22

12

22

=

maize (2 )

40919

27716

5148

51149.35

-

409.19

277.16

77.22

51.48

277.16

1

sorghum (2 )

27716.26

pepper (2 )³

10296.00

77.22

51.48

277.16

-

4

chickpea (2 )

5148

27716

-

5

bean (2 )

207.87

6

Similarly, the considered problem under the three scenarios can also be

solved for pessimistic and mixed DMs. The compromise solutions for each

scenario under pessimistic and mixed viewpoints are presented in the upper

and lower parts of Table 18, respectively.

Table 18: Solutions under pessimistic and mixed perspectives.

Scenario

ꢃ

-

Z

1

2

3

1

2

3

0.40

0.30

0.40

0.30

(168.27, 0.00, 304.30, 84.61, 153.51, 304.30)

(144.67, 0.00, 661.32, 141.05, 47.74, 67.97)

(412.14, 0.00, 277.56, 47.74, 47.74, 277.56)

(189.12, 0.00, 317.99, 72.76, 117.14, 317.99)

(144.67, 0.00, 661.32, 104.06, 47.74, 104.95)

(414.56, 0.00, 276.82, 46.80, 46.80, 276.81)

0.4132

0.4626

0.4495

0.4067

0.4352

0.4491

Based on the obtained results, the arable land allocated to the six crops

under the three approaches regarding the conventional pattern and the

three scenarios is presented in Figure 5.

(a)

(b)

(c)

Figure 5: Allocated land area to the six crops under the optimistic (a), pessimistic (b) and mixed (c) perspectives relative to the existing pattern.

Let - ̲, - ̲, and -

̲

denote the compromised solutions for the ﬁrst,

non-membership degrees of the objectives for the pessimistic and mixed

variants of the problem, are presented in Table 19.

1

3

second, an²d third scenarios, respectively, under pessimistic and mixed

views (Table 18). The objective values, along with the membership and

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

Table 19: Goal achievement under pessimistic and mixed perspectives.

Pessimistic perspective

Mixed perspective

5 (- ̲)

1

4

1

4

1

5(X₁̲)

27724¹.77

36²105370

28³462070

55064

28719¹.63

34²095870

28³677430

55719

ꢀ

ꢁ

0.44

0.45

0.38

0.34

0.54

0.40

0.35

0.31

0.00

0.02

5 (- ̲)

1

2

4

2

1

2

4

2

5(X₂̲)

28745².80

46²965540

30³707130

57780

0.52

29115².66

40²433450

30³115760

55931

0.58

ꢀ

ꢁ

0.78

0.34

0.86

0.25

0.41

0.00

0.21

0.06

0.00

5 (- ̲)

1

3

4

3

1

3

4

3

5(X₃̲)

37470³.45

36²717860

28³917350

66314

37538³.86

36²566940

28³909570

66329

ꢀ

ꢁ

0.42

0.00

0.68

0.35

0.00

0.38

0.42

0.00

0.67

0.35

0.38

0.00

Based on the obtained solutions using the pessimistic and mixed approaches (Table 18), the agricultural resource consumptions for the three scenarios

under each perspective, in order, are presented in Tables 20 and 21.

Table 20: Resource consumption under the pessimistic viewpoint.

, (- ̲)

1

2

1

6

1

7

1

8

1

31069¹67

101500

68³386.55

5⁴84.73

7⁵00.80

1057.30

2245.60

1015.00

, (- ̲)

1

2

3

2

6

2

7

2

8

2

36215²77

106274

112424.80

3⁴27.28

8⁵56.96

1133.26

2254.39

1015

, (- ̲)

1

2

3

6

3

7

3

8

3

33922³48

106274

88³821.33

3⁴96.91

8⁵97.87

1086.61

2433.60

1015

Table 21: Resource consumption under the mixed viewpoint.

, (- ̲)

1

2

1

6

1

7

1

8

1

31102¹09

101500

69³990.82

5⁴44.27

7⁵45.60

1051.38

2235.20

1015

, (- ̲)

1

2

3

2

6

2

7

2

8

2

35330²70

106274

105027.60

3⁴08.79

8⁵84.70

1114.77

2254.39

1015

, (- ̲)

1

2

3

6

3

7

3

8

3

33899³83

106180.50

88³862.42

3⁴93.83

8⁵98.99

1085.21

2432.72

1015

The OCP obtained using the proposed optimization method was compared

with existing patterns. Since the second scenario represents a commonly

practiced farming system in the district, a comparison was made between

the existing cropping pattern and the optimized plan under the optimistic

approach for this scenario.

29, 470.15@C; of yield was obtained under the existing cropping pattern,

but this is slightly reduced to 29, 358.32@C; in the proposed approach. If

we consider the remaining objectives, the total gain was 41, 510, 180ꢂ)ꢅ in

the existing pattern, but it can be increased to 44, 088, 180ꢂ)ꢅ applying the

proposed farming pattern. On the other hand, the cost of cultivation and

labor force were 31, 413, 470ꢂ)ꢅ and 64, 160<3, respectively, in the existing

farm plan, and these can be minimized to 30, 558, 420ꢂ)ꢅ and 57, 384<3,

respectively, by employing the suggested farming pattern.

As shown in Table 9, the area of land allocated to maize, teﬀ, chickpea and

bean in the existing situation was 221.03ℎ0, 141.41ℎ0, 62.29ℎ0, and 121.57,

respectively, while in the proposed pattern, as presented in Table 14, the

land area allocated to these crops respectively decreased to 156ℎ0, 0.00ℎ0,

51.45ℎ0, and 74.57. The area of land allocated to sorghum and pepper in

the existing situation was 401.33ℎ0 and 106.42ℎ0, respectively, while in

the proposed pattern, the land area allocated to these crops respectively

increased to 661.58ℎ0 and 122.85.

Employing the proposed cropping pattern, the objectives are achieved

with higher degrees of membership and lower degrees of non-membership

compared to the existing cropping plan, except for the maximization of the

yield target. However, in the existing situation, the production objective is

accomplished at the cost of agrarian assets and the remaining goals.

There are notable diﬀerences between the conventional farming system

and the suggested farming plan in the usage of manpower, fertilizers,

and agricultural machines; whereas there are slight diﬀerences in the

usage of herbicides and pesticides. If we consider the fertilizer constraint,

208, 591.20:, of fertilizer was required to cultivate 1, 054ℎ0 of farmland

in the existing farming pattern, while the suggested farming plan requires

216, 876:, of fertilizer to cultivate 1, 066.48ℎ0 of farmland within the two

production cycles. To cultivate the allocated crops on the respective areas

of land, 2, 392ℎAB (machine hrs.) were required using the existing plan,

whereas the suggested plan require 2, 272ℎAB.

In the conventional farming pattern, 991.76ℎ0 (96.00%) of 1, 033ℎ0 of land

is allocated to ﬁve crops in the ﬁrst round of farming, and 62.29ℎ0 (28.18%)

of 221.03ℎ0 of land is allocated to chickpea in the second round farming.

In the proposed cropping pattern, 1, 015ℎ0 (98.26%) of 1, 033ℎ0 of land is

allocated to four crops in the ﬁrst season of farming and 51.48ℎ0 (33%) of

156ℎ0 of land is allocated to chickpea in the second round of farming.

The results of the study showed that, including in the remaining scenarios

and DM approaches, teﬀ should not be included in the farming patterns

in favour of increasing the land area for other crops to attain better results

regarding all objectives and constraints.

Comparing the obtained results of optimistic approach under the ﬁrst,

second, and third scenarios, Tables 12, 14, and 16, respectively, the following

assessments have been made.

A comparatively wide area of land is allocated to sorghum and maize, under

From the allocated arable land in the existing farm patterns, a total of

Teferi H.T.,(2026)

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East Afr. J. Biophys. Comput. Sci. (2026), Vol. 7, Issue. 1, 43-63

the second and third scenarios, respectively, while pepper and bean are

allocated in a wide area of farmland under the second and third scenarios,

respectively. In contrast, Chickpeas share the largest area of land under

the ﬁrst scenario. Considering the resource constraints, water and fertilizer

consumption can be minimized by applying the ﬁrst farming scenario,

while herbicide and pesticide consumption can be reduced by applying

the second scenario. Whereas seed utilization can be minimized in the

second scenario. In the achievement of the considered objectives, enhanced

overall production and minimum expenditure are attained under the third

scenario. While the total gain is signiﬁcantly improved under the second

scenario and the number of manpower is suﬃciently minimized by applying

the ﬁrst farming scenario.

soil degradation unless resources are managed properly. All stakeholders

need to take action, especially in reducing fertilizer use. For example, using

manure and compost can help protect soil fertility, and practices like crop

rotation can lower the need for herbicides.

The study has numerous beneﬁts in assisting the managers of LSF and

farmers of the district for optimal management of agricultural resources. It

also indicates the advantages of OCP to overcome the potential disaster of

crops due to climate change and soil infertility. Furthermore, the study can

be used to predict promising cropping plans from a long-term perspective

as well.

A proﬁcient IFMOO model is proposed to address uncertainties and

associated risks of agriculture, aiming to achieve sustainable crop

production goals. Incorporating the risk management analysis model

into the IFMOO model can increase the eﬃciency and applicability of the

proposed approach to APP problems. Moreover, higher-order extensions of

IFO techniques, such as hesitant IFO (Teferi et al., 2025), are also helpful in

capturing the hesitation among DMs.

Based on the results of the three variants of the problem, as presented in

Tables 12, 14, 16, 18, 19, 20, and 21 there is signiﬁcant variation in the land

allocation to the six crops under each scenario. This proves the solution of

APP problem is contingent upon the DM’s perspective. For example, in the

ﬁrst scenario, the land allocated to maize under the optimistic approach

is reduced by 23.33ℎ0 and 2.48ℎ0, respectively, in the pessimistic and

mixed approaches. Whereas the cropland allocated to bean in the mixed

approach, respectively, decreased by 13.69ℎ0 and 47.09ℎ0 in the pessimistic

and optimistic approaches.

Data availability

The detailed experimental data used to support the ﬁndings of this study

are included in the supplementary information ﬁle(s).

Acknowledgements

The author gratefully acknowledges the Editor-in-Chief and the anonymous

reviewers for their careful evaluation, insightful comments, and

constructive suggestions. Their contributions have substantially improved

the presentation, clarity, and scientiﬁc rigor of this manuscript.

Conflict of Interest

The author declares that he has no competing ﬁnancial interests or personal

relationships that could have inﬂuenced the work reported in this paper.

The total cost of cultivation and manpower goals are better minimized

under a pessimistic approach, while total crop production and proﬁt are

better achieved under the mixed and optimistic approaches, respectively.In

the second scenario, a relatively equal large land area is allocated to

sorghum under the three approaches, whereas for beans, a wide area is

allocated under the mixed approach, but this is reduced by 30.38ha and

36.98ha under the optimistic and pessimistic approaches, respectively. The

farmland allocated to maize and chickpea remains the same under the

pessimistic and mixed approaches and diﬀers by 11.33ha for maize and

3.74ha for chickpea under the optimistic approach. In this scenario, the

yield maximization target is improved under the optimistic approach, while

the proﬁt maximization target is better achieved under the pessimistic

approach. In contrast, the cost and manpower minimization targets are

enhanced under the mixed approach.

Funding

The author did not receive any funding for this research.

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