Mathematical Programming Model for Multi-Objective Smallholder Farming Systems in North-East Nigeria

Farm households in North-east Nigeria (NEN) are mostly smallholders engaging predominantly in crop production. Provision of household food requirements is a vital objective in smallholder agriculture. However, its inclusion as a constraint in cropping patterns can affect farm-level returns associated with lower food crop value. A representative sample of 120 farmers was taken to model the farming system in NEN. The purpose was to evaluate the effect of food security as a constraint on farm income. Twenty (20) cropping activities were identified in the farming system: six sole cropping and fourteen inter-cropping patterns. Nine most popular cropping activities, practiced by 82.5% of the respondents, were selected as cropping activities for the model. The inter-cropping activities consist of food and cash crops. Farm level Mathematical programming techniques were constructed to model the farming system under choice constraints. First, a baseline model, the conventional linear programming (LP) technique, was constructed to model the system as a profit maximization problem. A combination of nine (9) most popular cropping activities and resources used were incorporated in the baseline model. The model selected Maize and Bambara nuts as the optimal activity to be cultivated on 1.3 ha with a profit of ₦734,763.73. The profit from the optimal activity in the baseline model was compared with that of the activity with the highest profit in the current plan; the result suggested a potential 21% increase in profit. Input resources such as land, fertilizer, and agrochemicals were not fully utilized as indicated in their respective slack values. Labour for processing and harvesting legumes was the binding resource with shadow prices in the optimal plan. This was due to capital constraints to hire. The second technique, however, presented a multi-objective programming (MOP) technique, with multiple model outputs to reflect farmers varying objectives such as profit maximization, food security objective, and their respective resource requirements. Two different solutions were obtained from this model: one with only maize as a food security constraint and the second with maize and beans as food security constraints. The estimated profit obtained from the two solutions in the MOLP final plans revealed 7% and 21% reduction in net profit, respectively, relative to the optimal solution obtained in the baseline profitability model. Therefore, this study recommends the extension of both LP and MOP results for consideration as a flexible decision-making support tool for the farming system in the study area. In the sensitivity analysis, the interaction between input (credit) and output was demonstrated so that intended decision-makers could understand the response of credit to farm profit. It is therefore recommended that government and non-government agencies should structure accessible and affordable sources of farm finance for farmers in the study area.

Introduction

Typical sub-Saharan smallholder farmers depend largely on farm activities not only for subsistence food but also as a source of income [1]. Hence, the drive for profit is embedded in their production decisions. In arable agriculture, with no livestock, these production decisions consist of i) the choice of crops to be grown; ii) the area of each crop; and iii) the allocation of scarce resources—land, labour, and capital-to crops grown [2]. Since resources and time are finite, smallholder farmers are conscious of the intrinsic challenges of how to allocate scarce production resources when making farm decisions [3]. These periodic farm decision processes influence how factors of production are transformed in a farming system to achieve desired objectives [2], [4]. This has underscored the need for enhanced planning tools to cope with the constrained nature of the decision-making processes in farm planning. These tools range from standard budgeting tools (for example, using a spreadsheet) to other computer-based programming modelling techniques, such as Linear Programming (LP), Multi-Objective Genetic Algorithm (MOGA), Dynamic Stochastic Programming (DSP), etc., that have been used in farm management analysis [5]–[8]. Bharwani et al. [9] however, advocate the use of mathematical programming model such as Linear Programming (LP) as appropriate farm management technique for capturing production decisions making problems and the financial implication of each decision taken by farmers. Linear programming is a technique that depicts complex relationships through linear functions and then finds the optimum point of interest [10], [11]. The development of the linear programming (LP) technique has been applauded as one of the most important scientific advances of the mid-20^th century. Its application has been universally accepted across the field of science [12]. The application of Linear programming (LP), as a farm decision support mechanism presents a simple and systematic method of determining mathematically the optimum farm plan for the selection and combination of farm enterprises and resources to achieve an optimal objective [13].

Several studies have employed LP modelling technique with empirical evidence to justify its effectiveness as a decision support tool for determination of profit maximising farm plan [3], [6], [8], [14]–[17]. However, LP model with single profit maximization objective function is oversimplified, as in many cases, intended users are diverse with varying objectives [18]. Nonetheless, there has been a possible extension of LP to incorporate more than one objective that supports the decision-making of managers with multi-objective targets [19]. In subsistence agriculture, the provision of household food is a key objective of smallholder cropping decisions, with income pursuit as another driver. Hence, Rehman and Romero [20], their work extended the frontiers of LP and encouraged its application such that it could handle multiple criteria in the decision-making process of a farming system. Several other authors in the literature [21]–[23] have also considered multiple objectives of farmers in modelling their investigation. Overall, farmers do not want to be exposed to the risk of family food shortage and, at the same time, need some income for the family, and these influence their farm choices and resource usage [24]. However, to the author’s knowledge, there is limited or no availability of literature that investigates multi-objective mathematical modelling of smallholder farming systems in North-east Nigeria (NEN).

Hence, the specific aims of the study are: i) to understand the current farming system characteristics in NEN– cropping pattern, resource use, constraints, profitability, ii) to identify optimal solutions using the conventional LP model under the system characteristics recognized in i), and iii) to reveal scenario solutions with household food security as a constraint using MOP-LP variant, and its impact on optimal solutions iv) to analyse the impact of credit on profitability.

Materials and Methods

Study Area

The study was carried out in the North-East Nigeria (NEN) between 2019 and 2021. Nigeria consists of six geo-political zones, and each zone is made up of six state Governments. The Northeastern region consists of six states: Adamawa, Borno, Bauchi, Gombe, Taraba, and Yobe. The specific study areas are Michika Local Government Area and Mubi South Local Government areas of Adamawa State, North-East Nigeria. Michika lies between latitude 10′ 37′ 7.65″ N and longitude 13° 24′ 4.25″ E, and Mubi South lies between 10° 15’ 48.76” N and 13°16′ 10.66″ E (Google Earth pro-2022). Michika shares a boundary with Lassa and Madagali Local Government to the North, Mubi North Local Government to the south, and Cameroon Republic to the east. Mubi South LGA shares a boundary with the Cameroon Republic to the east, Mubi North to the north, Hong LGA to the west, and Maiha and Fufore LGAs to the south. The dominant occupation of the people in the study area of Nigeria is farming, and the major crops grown are maize, cowpea, groundnuts, Bambara nuts, sorghum, and rice [17]. The climate in the region is yearly warm or hot with an average temperature of 35°C [25].

Sampling and Sampling Technique

Multistage sampling was used to achieve 120 complete responses from smallholder farmers in this study region. The first stage was the deliberate selection of Adamawa state from the region based on its relative accessibility to the researcher and the diversity of the farming communities. The second stage of the sampling technique was the selection of two local government areas (LGAs) in the state, those of Michika and Mubi South, which hold records of registered farmers, facilitating survey dissemination. Next was the purposive selection of ten villages with an equal split between Michika and Mubi South Local Government Areas. The purposive selection of these villages was due to the predominance of farm concentration in those villages, and they are also motorable and non-mountainous villages like others. The record list of farmers was obtained from the cooperative societies and Agricultural Development Offices (ADP). Primary data was collated with the aid of a structured questionnaire over a three-year period.

Analytical Tool

For the analysis, descriptive statistics were used to present the farming system characteristics, while the conventional LP and multi-objective LP techniques were used to model the current farming systems for the identification of optimal solutions and multi-objective solutions within constrained resources. Parametric programming (post-optimality) was used to analyze the impact of credit on farm profitability.

Specification of the Models

The mathematical expression of the conventional LP model constructed for this objective as adopted by Al-Nassr [26] and is specified below:

(1) $$\begin{array}{r}MaxZ={\sum }_{j=1}^n{P}_j{X}_J={\sum }_{i=1}^n{\sum }_{j=1}^m{C}_{ij}{X}_{ij}\end{array}$$

Subject to

(2) $$\begin{array}{r}{\sum }_{j=1}^n{a}_{iJ}{x}_j(\le )\text{bi}\end{array}$$

(3) $$\begin{array}{r}nonnegativitycontraints=X_j\ge 0\end{array}$$

where i is constraints to 1, 2, 3…m.

The multi-objective technique by Kelechi and Christian [27] was modified and adopted. The objective functions for this MOP included the attainment of minimum household food requirements in bulk weight (kg) for staple foods, namely, Maize and Beans. This is mathematically expressed as a modification of (1):

(4) $$\begin{array}{r}MOP=\left(f/z\right)\{\begin{array}{ll}\sum _{i=1}^n{\sum }_{j=1}^m{C}_{ij}{X}_{ij}& {\sum }_{j=1}^n{a}_{iJ}{x}_j(\le )\text{b}\\ {\sum }_{j=1}^n{f}_k{x}_j& {\sum }_{j=1}^n{f}_k{x}_j\ge \text{F}{F}_{kmb}\end{array}\end{array}$$

to:

(5) $$\begin{array}{rcl}& & Minimumannualquantityofmaizeinkg\\ & & requiredforhouseholdconsumption\\ & & ={\sum }_{j=1}^n{f}_k{x}_j\ge F_{km}\end{array}$$

(6) $$\begin{array}{rcl}& & Minimumannualquantityofbeansinkg\\ & & requiredforhouseholdconsumption\\ & & ={\sum }_{j=1}^n{f}_k{x}_j\ge F_{kb}\end{array}$$

where Z denotes profit, f denotes food for household, $X_j$ denotes the decision variable eg land size or number of labour days allocated to an enterprise, $P_j$ denotes the total value per ha from jth enterprise, $C_{ij}$ denotes the cost per unit of ith resource use in jth activity, $X_{ij}$ denotes the amount of ith input used in jth activity, $a_{ij}$ denotes the amount “a” of “i” resource used in the production of one unit of “j” activity, $b_i$ denotes the available level of resource I where i = 1.2. N, m denotes the number of activities in the model, n denotes the number pf resources in the model.

Results

The Farming System Characteristics

As presented in Fig. 1, a total of 20 cropping patterns were identified: 14 were intercropping enterprises, 76% of the farmers practiced this inter-cropping, and 88% of the respondents cultivated food crops either as sole or in combination with other crops. The finding from the study reveals Maize and beans as the staple food in the region, as reported by 78% and 81% of respondents, respectively, and the quantity required by the average family of six was 433 kg and 76 kg of maize, respectively.

Fig. 1. Distribution of farmers by Enterprise selection.

LP Model Results

Current Best Plan and LP Model Optimal Solution at Observed Resource Levels

Table I presents the optimal activity suggested with a profit of ₦ 734,764, which represented an increase of 21% profit over the prevailing plan with the highest profit. The current plan could not enter the model; the optimal activity selected was Maize and Bambara nut to be cultivated at 1.31 ha. Table II was the sensitivity report in the final plan with activity costs.

Table I. Current and Optimal Enterprise Levels at Observed Resource Level in the Baseline Model

Plan	Enterprise	Optimum level (ha)	Credit (₦)	Profit (₦)	Increase/decrease over current plan. ₦	Increase/decrease (%)
Current plan	Sorghum & bambara nut	1.31	200,000	607,837	126,926.33	21%
Optimal plan	Maize & bambara nut	1.31	200,000	734,764

Table II. Sensitivity Report in the Final Plan with Activity Costs

Enterprise	Final value	Reduced cost	Objective coefficient
Maize	0	−311,210.74	164,060.4
Beans	0	−93,455.97	363,710.1
Maize & G/nuts	0	−243,510.72	424,723.16
Maize & beans	0	−117,527.31	484,626.6
Maize & sorghum	0	−213,595.28	261,094.3
Maize & B/nut	1.306324856	0	601,487.2
G/nut & sorghum	0	−122,612.98	544,220.7
Beans & sorghum	0	−83,943.71	458,761.9
Sorghum & B/nut	0	−59,868.10	607,837.4

Sensitivity Report in the Final Plan with Activity Costs

The sensitivity analysis of the final plan displays the final values of the optimal activity and the negative cost of excluded sub-optimal activities in terms of their various reduced costs. Hence, the optimal activity has zero reduced cost.

Optimal Resource Levels in The Optimal Plan

Table III shows the different allocations of resources in the model. Credit, labor for harvesting legumes, and labor processing legumes were the binding resources in the optimal plan.

Table III. Resource Use and their Corresponding Status in the Final Plan

Resource id	Used	Slack (unused)	Status	Shadow price
LAND	1.3	0.69	Not binding	0
NPKKGHA	304.4	695.63	Not binding	0
UREAKGHA	101.9	398.11	Not binding	0
SSPKGHA	0.0	250	Not binding	0
FORCE UP	2.6	47.39	Not binding	0
ATRA FORCE	0.0	50	Not binding	0
PARAE FORCE	2.6	47.39	Not binding	0
SLASHER	2.6	47.39	Not binding	0
LARAFORCE	2.6	47.39	Not binding	0
LAMDA PLUS	1.3	48.69	Not binding	0
MAGIC FORCE	2.6	47.39	Not binding	0
CATERPILLAR FORCE	0.0	50	Not binding	0
LANDPREPLAB	5.2	18.77	Not binding	0
PLANTING CEREALS	10.5	29.55	Not binding	0
PLANTING LEGUMES	13.1	46.94	Not binding	0
WEEDING LAB	17.0	133.02	Not binding	0
CHEMAPPLAB	3.9	16.08	Not binding	0
HARVESTING CEREALS	13.1	66.94	Not binding	0
HARVESTING LEGUMES	16.0	0	Binding	2200
PROCESSING CEREALS	7.8	32.16	Not binding	0
BUY CREEDIT	161000.0	0	Binding	1.98549509
CREDIT LIMIT	200000.0	0	Binding	1.83549509
PROCESSING LEGUMES	8.0	0	Binding	1600

MOP Model Result with Minimum Household Food Requirement

Minimum household food requirements were incorporated as constraint separately into the MOP model as a minimum food constraint. The financial effects of these inclusions were presented (Table IV) below as optimal solution 1 and optimal solution 2. The comparative analysis of these solutions with the optimal plan obtained from the baseline (profitability-driven) model illustrated the income risk/gain and trade-off associated with the choice of either of these three plans.

Table IV. Effects of Food Security Constraints on Profitability/Optimal Plan

Objective	Solution	Enterprises	Optimum level (ha)	Borrowing (₦)	Total gross margin (₦)	Decrease/increase over optimalplan (₦)	Decrease/increase (%)
Profitability	Optimal plan	Maize & B/Nut	1.3	200,000	734,763.73	Baseline	NIL
Food security (min maize)	Optimal solution 1	Maize & Beans	0.45	200,000	681,876.44	−52,887.29	−7%
		Maize & B/Nut	0.85
Food security (min maize & beans)	Optimal solution 2	Maize	0.45	200,000	581,635.06	−153,128.67	−21%
		Beans	0.14
		Maize & B/Nut	0.82

Parametric Effect of Increased Farm Finance on The Optimal Plan

This study has identified capital as a key resource with binding constrained effects on farm productivity. A parametric test was therefore performed and tested by varying the value of the capital as a constraint resource (Table V). Capital was increased by 100%, and the result indicated an increase in profit of 45% relative to the previous profit in the optimal plan (baseline). This has resulted in the utilization of other slack resources.

Table V. The Effect Increased Capital on the Optimal Plan

	Capital required (N)	Optimal level (ha)	Gross margin (N)	Increase (N)	Increase (%)
Optimal plan (latter)	400,000	1.3	1,065,820.06	331,056.33	45%
Optimal plan (former)	200,000	2	734,763.73

Discussion and Conclusion

In the analysis of the model result, the optimal plan shows an increase in profit of 21% over the existing plan. All the other activities in the model, except the optimal plan, would have a reduced cost effect if forced into the plan, as seen in the sensitivity report. Optimal activity was the only enterprise with zero reduced cost.

In the resource use model, capital and labour for processing and harvesting of legumes were the only constraints with zero slack values. This means an additional one unit of any binding resource will improve optimal value with the shadow price equivalent. Hence, additional use of non-binding resources has no positive impact on net profit. For instance, an additional ₦ 1 credit will improve the optimal value by ₦ 0.98K. Economic gains were not the only factors driving smallholder farmers’ cropping decisions in the study area; household food security was also a very critical factor. The MOP model reflects the reality of this farming system, in which farmers have food security constraints limiting their choice for absolute profit-maximising enterprises. In ensuring average household maize for family consumption, the farmer must allocate 0.45 ha of land for maize cultivation, which decreases the optimal value by 7%.

Similarly, if the farmer is to ensure the average for household food of maize and beans, then 0.45 ha and 0.14 ha must be allocated, respectively, in which case the expected gross margin decreases by 21% relative to the optimal plan. Sensitivity analysis is important because the real problems faced by farmers are associated with the changing world environment [28]. Sensitivity options broaden the choices available to decision-makers. Hence, the capital was increased by 100%, and the result indicated an increase in profit by 45% relative to the profit in the optimal plan (baseline).

In conclusion, this study highlights the relevance of improved farm management techniques in rural development. This study concludes that enterprise selection and farm resource allocation by smallholder farmers in the study area were not optimal. The model outcome indicates that farmers in the study region are able to improve productivity even with current resources available. Hence, the model template is suggested for adoption by farmers in the farming system to aid in farm decision-making processes. The study also revealed the Limiting effect of food security constraints by smallholder farmers on farm gross margin. This calls for increased socio-economic activities for increased employment that can boost off-farm income. The importance of capital investment in rural agriculture was also emphasized by the potential impact it could have on income and livelihood. It is recommended that the government and other relevant agencies make affordable and accessible agricultural financing a priority in developing the farming system in the region.