Simulation and mathematical programming decision-making support for smallholder farming

Abstract

Many mathematical programs have been developed over the past 50 years to aid agricultural experts and other farming decision-makers. The application of these mathematical programs has seen limited success because their development has focused on mathematical theory as opposed to the requirements needed for application. This paper describes the development of two mathematical programs that were designed to integrate with a visualization simulation that aids a nontraditional group of agricultural decision-makers: illiterate Sri Lankan subsistence farmers. The simulation was designed to help these illiterate farmers make business decisions about their crop selection choices which, in turn, will help them develop their business plans required for obtaining bank micro-loans. This paper’s focus is on the use of linear programming as a potential tool to demonstrate the benefits of crop diversification and rotation to the farmer based on various available crop types. It also highlights the issues using such an approach.

Appendix: program formulation

The complete mathematical formulations for linear and quadratic program are given in these appendices without comment as a holistic formulation. The original equation numbering has been used to help the reader refer back to the commentary in the main text if need be. The complete list of variables is given first:

$x_{i}^{a,t}$ :

Binary indicator of whether crop type “i” was planted in field “a” in session “t”

$f_{i}^{a,t}$ :

Binary indicator of whether fertilizer type “i” was used in field “a” in session “t”

$b_{i}^{a,t}$ :

Binary indicator of whether pesticide type “i” was used in field “a” in session “t”

$x_{i}$ :

Crop of type “i”

$f_{i}$ :

Fertilizer of type “i”

$b_{i}$ :

Pesticide of type “i”

$a$ :

Field indicator

$t$ :

Time indicator

$T$ :

Total sessions considered (= 20)

$N_{a}$ :

Number of fields (= 4)

$N_{x}$ :

Number of crop types (= 4)

$N_{f}$ :

Number of fertilizer types (= 2)

$N_{b}$ :

Number of pesticide types (= 2)

$R\left( F \right)$ :

Yield bonus from the previous year if field was left fallow

$c\left( . \right)$ :

Cost of product of either crop seed or chemical

$Y\left( {x_{i} } \right)$ :

Yield, in monetary value, from crop type “i” per field per session

$R\left( {x_{j} ,x_{i} } \right)$ :

Crop rotation impact from planting crop “j” the year before planting crop “i,” minus the yield bonus from the previous year field being fallow

$F\left( {f_{i} } \right)$ :

Yield impact from using fertilizer type “i”

$B\left( {b_{i} } \right)$ :

Yield impact from using pesticide type “i”

1.1 Linear program formulation

The linear program formulation is given in this appendix:

${\text{Maximize}}\mathop \sum \limits_{t = 1}^{T} \left( {Y_{t} - c_{t} } \right)$ (1).

s.t.

$$ c_{t} = \mathop \sum \limits_{a = 1}^{{N_{a} }} \left( {\mathop \sum \limits_{i = 1}^{{N_{x} }} c\left( {x_{i} } \right)x_{i}^{a,t} + \mathop \sum \limits_{i = 1}^{{N_{f} }} c\left( {f_{i} } \right)f_{i}^{a,t} + \mathop \sum \limits_{i = 1}^{{N_{b} }} c\left( {b_{i} } \right)b_{i}^{a,t} } \right) \quad\forall t $$

(2)

$$ \begin{aligned} Y_{t} & = \sum\limits_{i = 1}^{{N_{x} }} {Y\left( {x_{i} } \right)} \sum\limits_{a = 1}^{{N_{a} }} {\left( {x_{i}^{a,t} \left( {1 + R\left( F \right)} \right) + \sum\limits_{j = 1}^{{N_{x} }} {R\left( {x_{j} ,x_{i} } \right)I\left( {x_{j}^{a,t - 1} ,x_{i}^{a,t} } \right)} + \sum\limits_{j = 1}^{{N_{f} }} {F\left( {f_{j} } \right)I\left( {f_{j}^{a,t} ,x_{i}^{a,t} } \right)} } \right.} \\ & \quad + \left. {\sum\limits_{j = 1}^{{N_{f} }} {\left( {f_{j} } \right)I\left( {f_{j}^{a,t} ,x_{i}^{a,t} } \right)} + \sum\limits_{k = 1}^{{N_{b} }} {B\left( {b_{k} } \right)I\left( {b_{k}^{a,t} ,x_{i}^{a,t} } \right)} } \right)\quad\forall t > 0 \\ \end{aligned} $$

(3)

$$ m_{0} = 100,000 $$

(4)

$$ x_{i}^{a,t} + x_{j}^{a,t - 1} - 2.I\left( {x_{i}^{a,t} ,x_{j}^{a,t - 1} } \right) \le 1 \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\}, \quad\forall j \in \{ 1,2, \ldots ,N_{x} \} ,\quad\forall a,\quad\forall t $$

(5)

$$ x_{i}^{a,t} + x_{j}^{a,t - 1} - 2.I\left( {x_{i}^{a,t} ,x_{j}^{a,t - 1} } \right) \ge 0 \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad\forall j \in \{ 1,2, \ldots ,N_{x} \} ,\quad\forall a,\quad\forall t $$

(6)

$$ x_{i}^{a,t} + f_{j}^{a,t} - 2.I\left( {f_{j}^{a,t} ,x_{i}^{a,t} } \right) \le 1 \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\}, \quad\forall j \in \{ 1,2, \ldots ,N_{f} \} ,\quad\forall a,\quad\forall t $$

(7)

$$ x_{i}^{a,t} + f_{j}^{a,t} - 2.I\left( {f_{j}^{a,t} ,x_{i}^{a,t} } \right) \ge 0 \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad \forall j \in \{ 1,2, \ldots ,N_{f} \} ,\quad\forall a,\quad\forall t $$

(8)

$$ x_{j}^{a,t} + b_{k}^{a,t} - 2.I\left( {b_{k}^{a,t} ,x_{j}^{a,t} } \right) \le 1 \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\}, \quad\forall k \in \{ 1,2, \ldots ,N_{b} \} ,\quad\forall a,\quad\forall t $$

(9)

$$ x_{j}^{a,t} + b_{k}^{a,t} - 2.I\left( {b_{i}^{a,t} ,x_{j}^{a,t} } \right) \ge 0 \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad\forall k \in \{ 1,2, \ldots ,N_{b} \} ,\quad\forall a,\quad\forall t $$

(10)

$$ \mathop \sum \limits_{i = 1}^{t - 1} Y_{i} + m_{0} \ge \mathop \sum \limits_{i = 1}^{t} c_{i} \quad\forall t $$

(11)

$$ \mathop \sum \limits_{i = 1}^{{N_{x} }} x_{i}^{a,t} \le 1 \quad\forall a,\quad\forall t $$

(12)

$$ \mathop \sum \limits_{j = 1}^{{N_{f} }} f_{j}^{a,t} \le 1 \quad\forall a,\quad\forall t $$

(13)

$$ \mathop \sum \limits_{k = 1}^{{N_{b} }} b_{k}^{a,t} \le 1 \quad\forall a,\quad\forall t $$

(14)

$$ I\left( {x_{j}^{a,t - 1} ,x_{i}^{a,t} } \right) \in \left\{ {0,1} \right\} \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad\forall j \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad\forall a,\quad\forall t > 0 $$

$$ I\left( {f_{j}^{a,t} ,x_{i}^{a,t} } \right) \in \left\{ {0,1} \right\} \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad\forall j \in \{ 1,2, \ldots ,N_{f} \} ,\quad\forall a,\quad\forall t $$

$$ I\left( {b_{k}^{a,t} ,x_{i}^{a,t} } \right) \in \left\{ {0,1} \right\} \quad\forall i \in \left\{ {1,2, \ldots ,N_{x} } \right\},\quad\forall k \in \{ 1,2, \ldots ,N_{b} \} ,\quad\forall a,\quad\forall t $$

$$ t \in \left\{ {1,2, \ldots ,T} \right\} $$

$$ a \in \left\{ {1,2, \ldots ,N_{a} } \right\} $$

$$ x_{i}^{a,t} ,f_{i}^{a,t} ,b_{i}^{a,t} \in \left\{ {0, 1} \right\} \quad\forall i,\quad\forall a,\quad\forall t $$

$$ x_{i}^{a,0} = \left\{ {\begin{array}{*{20}c} 1 & {if} & {i = 0} \\ 0 & {if} & {o/w} \\ \end{array} } \right. $$

(15)

1.2 Quadratic program formulation

The complete quadratic program formulation is given in this appendix:

${\text{Maximize}}\mathop \sum \limits_{t = 1}^{T} \left( {Y_{t} - c_{t} } \right)$.

s.t.

$$ c_{t} = \mathop \sum \limits_{a = 1}^{{N_{a} }} \left( {\mathop \sum \limits_{i = 1}^{{N_{x} }} c\left( {x_{i} } \right)x_{i}^{a,t} + \mathop \sum \limits_{i = 1}^{{N_{f} }} c\left( {f_{i} } \right)f_{i}^{a,t} + \mathop \sum \limits_{i = 1}^{{N_{b} }} c\left( {b_{i} } \right)b_{i}^{a,t} } \right) \quad\forall t $$

$$Y_{t} = \mathop \sum \limits_{{i = 1}}^{{N_{x} }} Y\left( {x_{i} } \right)\mathop \sum \limits_{{a = 1}}^{{N_{a} }} x_{i}^{{a,t}} \left( {1 + R\left( F \right) + \mathop \sum \limits_{{j = 1}}^{{N_{x} }} R\left( {x_{j} ,x_{i} } \right)x_{j}^{{a,t - 1}} + \mathop \sum \limits_{{j = 1}}^{{N_{f} }} F\left( {f_{j} } \right)f_{j}^{{a,t}} + \mathop \sum \limits_{{k = 1}}^{{N_{b} }} B\left( {b_{k} } \right)b_{k}^{{a,t}} } \right)\quad\forall t > 0$$

$$ \mathop \sum \limits_{i = 1}^{{N_{x} }} x_{i}^{a,t} \le 1 \quad\forall a,\quad\forall t $$

$$ \mathop \sum \limits_{j = 1}^{{N_{f} }} f_{j}^{a,t} \le 1 \quad\forall a,\quad\forall t $$

$$ \mathop \sum \limits_{k = 1}^{{N_{b} }} b_{k}^{a,t} \le 1 \quad\forall a,\quad\forall t $$

$$ m_{0} = 100,000 $$

$$ t \in \left\{ {1,2, \ldots ,T} \right\} $$

$$ a \in \left\{ {1,2, \ldots ,N_{a} } \right\} $$

$$ x_{i}^{a,t} ,f_{i}^{a,t} ,b_{i}^{a,t} \in \left[ {0,1} \right] \quad\forall i,\quad\forall a,\quad\forall t $$

$$ x_{i}^{a,0} = \left\{ {\begin{array}{*{20}c} 1 & {if} & {i = 0} \\ 0 & {if} & {o/w} \\ \end{array} } \right. $$