ACP Mathematical Model with Economic Factors for an Existing Irrigation Scheme

The enhanced Best Performance Algorithm for the Annual Crop Planning Problem Based on Economic Factors

3.6 ACP Mathematical Model with Economic Factors for an Existing Irrigation Scheme

This section presents the enhancement of the ACP mathematical model. This model includes the market economic factors of the economy of scale, and the demand and supply relations. Explanations on the foundational ACP mathematical models can be found in Chetty and Adewumi (2013b, 2013c, 2013d, 2014). The mathematical model in this study relates to that of an existing irrigation scheme.

To implement the economy of scale influence, a ‘fixed cost’ variable is introduced. Hence, production costs are now explicitly differentiated as being fixed and variable costs of production. A fixed cost factor associated with the production of each crop will encourage a higher quantity of produce as the unit cost will decrease. This will result in a higher profit earned per crop. However, this influence is challenged by the demand and supply relations, as higher yields beyond the equilibrium point will result in lower producer prices; this will equate to less profit earned per unit (and vice versa).

In this model, equilibrium price is represented in terms of hectare allocations. This is achieved by making use of either (or both) of the demand or supply relational equations. With gross profits earned being dependent on hectare allocations, it is now interesting that hectare allocations and gross profits are influenced by the economy of scale and the demand and supply relational factors. This introduces added complexity, yet allows for more scalable solutions.

The ACP mathematical model, which includes the market economic factors of the economy of scale and the demand and supply relations are as follows:

61 3.6.1 Mathematical Notations

Indices

 𝑘 – Plot types. (1 = single-crop plots; 2 = double-crop plots; 3 = triple-crop plots; etc.).

 𝑖 – Indicative of the crop groups that are grown in sequence of each other on the same farming plot of land within the year, on plot type 𝑘 (𝑖 = 1 indicates the 1^st crop group; 𝑖 = 2 indicates the 2^nd crop group; 𝑖 = 3 indicates the 3^rd crop group; etc.).

 𝑗 – Indicative of the individual crops belonging to crop group 𝑖, on plot 𝑘.

Input Parameters

 𝑙 – Number of different farming plot types.

 𝑁_𝑘 – Number of sequential crop groups cultivated on plot 𝑘.

 𝑀_𝑘𝑖– Number of individual crops cultivated at stage 𝑖, on plot 𝑘.

 𝐻_𝑘𝑖𝑗 – Hectare allocation of crop 𝑗, at stage 𝑖, on plot 𝑘 as determined from the previous year.

 𝐿_𝑘𝑖 – Total area of land allocated for crop production at stage 𝑖.

 𝐹𝑅_𝑘𝑖𝑗 – Average fraction per hectare of crop 𝑗, at stage 𝑖, on plot 𝑘, which needs to be irrigated (1 = 100% coverage, 0 = 0% coverage).

 𝑅_𝑘𝑖𝑗 – Averaged rainfall estimates that fall during the growing months for crop 𝑗, at stage 𝑖, on plot 𝑘.

 𝐶𝑊𝑅_𝑘𝑖𝑗 – Crop water requirements of crop 𝑗, at stage 𝑖, on plot 𝑘.

 𝐴 – Volume of irrigated water that can be supplied per hectare (ha^-1).

 𝑃 – Price of irrigated water m^-3.

 𝑂_𝑘𝑖𝑗 – Operational cost ha^-1 of crop 𝑗, at stage 𝑖, on plot 𝑘. This cost excludes the cost of irrigated water per crop.

 𝐹_𝑘𝑖𝑗 – Fixed cost of production for crop 𝑗, at stage 𝑖, on plot 𝑘.

 𝑌𝐷_𝑘𝑖𝑗 – The expected yield in tons per hectare (t ha^-1) of crop 𝑗, at stage 𝑖, on plot 𝑘.

 𝑀𝑃_𝑘𝑖𝑗 – Producer price per ton of crop produced for crop 𝑗, at stage 𝑖, on plot 𝑘. This is the equilibrium price from the previous year of trading, at the hectares allocated. It is determined by the demand/supply relation.

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 𝐿𝑏_𝑘𝑖𝑗 – Lower bound of crop 𝑗, at stage 𝑖, on plot 𝑘. This reflects the minimum expected market demand in order to meet supply needs. This should to be determined by the producers.

 𝑈𝑏_𝑘𝑖𝑗 – Upper bound of crop 𝑗, at stage 𝑖, on plot 𝑘. This reflects the maximum expected market demand. The producers should also determine this.

Calculated Parameters

 𝑇𝐴 – Total volume of irrigated water that can be supplied to the total area of farming land within the year (𝑇𝐴 = 𝑇 ∗ 𝐴).

 𝐼𝑅 _𝑘𝑖𝑗 – Volume of irrigated water that should be supplied to crop 𝑗, at stage 𝑖, on plot 𝑘.

(𝐼𝑅_𝑘𝑖𝑗𝑚³ = (𝐶𝑊𝑅_𝑘𝑖𝑗𝑚 – 𝑅_𝑘𝑖𝑗𝑚) ∗ 10000𝑚² ∗ 𝐹𝑅_𝑘𝑖𝑗).

 𝐶_𝐼𝑅_𝑘𝑖𝑗 – The cost of irrigated water ha^-1 of crop 𝑗, at stage 𝑖, on plot 𝑘. (𝐶_𝐼𝑅_𝑘𝑖𝑗 = 𝐼𝑅_𝑘𝑖𝑗 ∗ 𝑃).

 𝐶_𝑘𝑖𝑗 – Variable cost ha^-1 of crop 𝑗, at stage 𝑖, on plot 𝑘. (𝐶_𝑘𝑖𝑗 = 𝑂_𝑘𝑖𝑗 + 𝐶_𝐼𝑅_𝑘𝑖𝑗).

Variables

 𝑋_𝑘𝑖𝑗 – Area of land, in hectares, that can be feasibly allocated for the production of crop 𝑗, at stage 𝑖, on plot 𝑘.

 𝐴𝑉_𝑘𝑖𝑗 – Average cost ha^-1 in considering the fixed and variable costs of production for crop 𝑗, at stage 𝑖, on plot 𝑘. (𝐴𝑉_𝑘𝑖𝑗 = (𝑋_𝑘𝑖𝑗𝐶_𝑘𝑖𝑗+ 𝐹_𝑘𝑖𝑗)/𝑋_𝑘𝑖𝑗).

 𝐸𝑃_𝑘𝑖𝑗 – Equilibrium price that is substituted by using either the demand or supply relations, which has dependency on 𝑋_𝑘𝑖𝑗 (e.g. Demand relation: 𝑋_𝑘𝑖𝑗(D) = a + b𝐸𝑃_𝑘𝑖𝑗; Supply relation: 𝑋_𝑘𝑖𝑗(S) = c + d𝐸𝑃_𝑘𝑖𝑗 where a, b, c and d are constants).

3.6.2 Optimization Model

Objective Function Maximize 𝑓 =

∑ ∑ ∑ 𝑋_𝑘𝑖𝑗(𝐸𝑃_𝑘𝑖𝑗∗ 𝑌𝐷 − 𝐴𝑉_𝑘𝑖𝑗)

𝑀_𝑘𝑖

𝑗=1 𝑁_𝑘

𝑖=1 𝑙

𝑘=1

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= ∑ ∑ ∑ 𝑋_𝑘𝑖𝑗(𝐸𝑃_𝑘𝑖𝑗∗ 𝑌𝐷 − 𝐶_𝑘𝑖𝑗) − 𝐹_𝑘𝑖𝑗

𝑀_𝑘𝑖

𝑗=1 𝑁_𝑘

𝑖=1

(3.7)

𝑙

𝑘=1

Equation 3.7 gives the objective function. The fixed cost variable 𝐹_𝑘𝑖𝑗 implements the economy of scale influence. The equilibrium price variable 𝐸𝑃_𝑘𝑖𝑗 (substituted in terms of hectare allocations, by using either of the demand or supply relational equations) are used to implement the market demand or supply influence. The constraints to the problem remain the same as found in Chetty and Adewumi (2013b, 2014).

Land Allocation Constraints

All solutions must satisfy the lower and upper bounds of each crop.

𝐿𝑏_𝑘𝑖𝑗≤ 𝑋_𝑘𝑖𝑗 ≤ 𝑈𝑏_𝑘𝑖𝑗 ∀𝑘, 𝑖, 𝑗 (3.8)

The summation of the land allocated for each crop 𝑗, at stage 𝑖, on plot 𝑘, must not exceed the total area of land available for crop production at stage 𝑖, on plot 𝑘.

∑ 𝑋_𝑘𝑖𝑗 ≤ 𝐿_𝑘𝑖 ∀𝑘, 𝑖 (3.9)

𝑀_𝑘𝑖

𝑗

Irrigated Water Constraints

The summation of the volume of irrigated water allocated to each crop must be less than the total volume that can be supplied to the irrigation scheme within the year.

∑ ∑ ∑ 𝐼𝑅_𝑘𝑖𝑗 ≤ 𝑇𝐴 (3.10)

𝑗 𝑖 𝑘

64 Non-negative Constraints

Arbitrarily, the lower and upper bound settings as well as the gross profits earned per crop must be non-negative.

𝐿𝑏_𝑘𝑖𝑗, 𝑈𝑏_𝑘𝑖𝑗, (𝐸𝑃_𝑘𝑖𝑗∗ 𝑌𝐷 − 𝐴𝑉_𝑘𝑖𝑗) > 0 ∀𝑘, 𝑖, 𝑗 (3.11)

3.7 Summary of, “On the Performance of new Local Search Heuristics