Linear Programming Solutions for Improving Wheat Delivery

Introduction

This assignment aims to use Linear Programming (LP) techniques to solve a real-world agricultural optimisation problem. To be precise, the assignment entails modelling the distribution of wheat deliveries between two farms over a planning horizon of three months. The issue is to strike a balance between land, yield, harvest capacity, labour and transportation costs versus the changing demand and wheat prices. The assignment will enhance problem-solving and report-writing abilities, which will be critical in more elaborate assessments.

The context of the problem is the supply chain of wheat in the months of September, October, and November. The demand and prices during these three months vary greatly: September demand is 1,800 tonnes at a price of $455 per tonne; October demand is 1,850 tonnes with a peak of $472 per tonne; and November demand is 1,750 tonnes at $465 per tonne. These changes give both threats and prospects. Although the increase in prices in October will encourage shifting supply, operational constraints reduce flexibility.

Linear Programming is especially applicable to this problem because it offers a systematic model to maximise profit with a variety of constraints. Defining decision variables, an objective function, and constraints make sure that optimal solutions can be determined systematically using LP. The tool that was employed to model and compute the solution was Excel Solver.

The deliverables for this project include:

1. A working Excel Solver model that demonstrates the LP solutions for baseline and silo scenarios
2. A written report that explains the model formulation, presents the results, and provides recommendations

Problem Description

The farmer operates two farms: Farm 1 (1,000 ha) and Farm 2 (950 ha). Wheat yield is estimated at 2.6 t/ha, which constrains maximum production. Monthly harvesting capacity is capped at 900 t per farm, representing the operational limit of machinery and labour. The farmer faces costs from sowing and growing ($375.84/ha), harvest labour ($29–30/ha), and freight ($6.5–7.2/t depending on farm and month). On the revenue side, wheat prices vary monthly, as noted earlier.

An extended option introduces silo storage. This allows wheat harvested in September to be stored and sold in October or November. Storage, however, incurs costs of $0.07/t for loading, $0.07/t per month for maintenance, and $0.07/t for unloading — totalling $0.21/t for one month of carryover. Additionally, silo capacity requires a capital investment of $300 per tonne of capacity.

The primary objective is to maximise profit, defined as total revenue minus sowing, growing, labour, freight, and storage costs. Importantly, total available supply (≈5,070 t) is less than total demand (5,400 t), meaning some demand will inevitably go unmet. Price volatility introduces risk, while the silo investment creates a trade-off between higher revenues and added costs.

Table 1. Key Input Data

Additional parameters: Farm 1 area = 1,000 ha; Farm 2 area = 950 ha; Sow & Grow cost = $375.84/ha; Silo CAPEX = $300/t

LP Model Formulation

Decision Variables

• xf,m: tonnes delivered from farm f ∈ 1,2 in month m ∈ Sep, Oct, Nov
• sSep→Oct, sSep→Novs: tonnes stored from September to October/November

Objective Function

Maximise Profit=m∑​Pm​(x1,m​+x2,m​)−[m∑​(F1,m​x1,m​+F2,m​x2,m​)+m∑​Yx1,m​​L1,m​+Yx2,m​​L2,m​+m∑​Yx1,m​+x2,m​​SG+Storage Costs]

Constraints

1. Demand balance: Deliveries plus storage inflows = monthly demand
2. Harvest cap: xf,m ≤ 900x
3. Land: ∑m (xf,m/Y) ≤ farm area
4. Flow balance (silo): Mills receipts = deliveries + stored inflows
5. Non-negativity: all variables ≥ 0