How to Optimize Monthly Oil Purchases and Production for Maximum Profit

Problem Description

Manufacturer needs to refine several raw oils and blend them to produce a sellable food product. Two categories of oils are required: vegetable oils (VEG 1, VEG 2) and non‑vegetable oils (OIL 1, OIL 2, OIL 3).

Spot and futures prices (USD/ton) for each oil by month (January–June) are given:

January: VEG1 110, VEG2 120, OIL1 130, OIL2 110, OIL3 115

February: VEG1 130, VEG2 130, OIL1 110, OIL2 90, OIL3 115

March: VEG1 110, VEG2 140, OIL1 130, OIL2 100, OIL3 95

April: VEG1 120, VEG2 110, OIL1 120, OIL2 120, OIL3 125

May: VEG1 100, VEG2 120, OIL1 150, OIL2 110, OIL3 105

June: VEG1 90, VEG2 100, OIL1 140, OIL2 80, OIL3 135

Additional considerations:

Final food product sells for $150 per ton.

Vegetable and non‑vegetable oils are refined on separate production lines.

Refining capacity is limited to 200 t of vegetable oil and 250 t of non‑vegetable oil per month.

No waste in refining; total refined oil equals total refined product.

Refining cost is negligible.

Storage limit per oil type is 1000 t; storage cost is $ ? per ton per month (value omitted).

Food product hardness must be between 3 and 6. Hardness is a linear blend of oil hardnesses: VEG1 = 8.8 VEG2 = 6.1 OIL1 = 2.0 OIL2 = 4.2 OIL3 = 5.0

Initial inventory: 500 t of each oil at the beginning of January; also 500 t of each oil must be on hand at the end of June.

Mathematical Model

Sets

T: set of months.

V: set of vegetable oils.

N: set of non‑vegetable oils.

O: set of all oil types (V ∪ N).

Parameters

p = 150 USD/t (selling price of final product).

s₀ᵢ: initial storage of oil i (500 t).

sᴛᵢ: target storage of oil i at end of horizon (500 t).

cᵢₜ: purchase price of oil i in month t (from the price tables).

hᵢ: hardness of oil i (values above).

γᵥ = 200 t (vegetable refining capacity per month).

γₙ = 250 t (non‑vegetable refining capacity per month).

H_min = 3, H_max = 6 (hardness limits).

k: storage cost per ton per month (not specified).

Decision Variables

xᵢₜ: amount of oil i purchased in month t.

yᵢₜ: amount of oil i refined (consumed) in month t.

zᵢₜ: inventory of oil i at end of month t.

qₜ: amount of final product produced in month t.

Objective

Maximize total profit over the planning horizon:

<code>∑ₜ (p·qₜ – ∑ᵢ cᵢₜ·xᵢₜ – k·zᵢₜ)</code>

Constraints

Inventory balance: zᵢ₍ₜ₋₁₎ + xᵢₜ = yᵢₜ + zᵢₜ for all i, t.

Refining capacity: ∑_{i∈V} yᵢₜ ≤ γᵥ and ∑_{i∈N} yᵢₜ ≤ γₙ for all t.

Hardness: H_min ≤ (∑ᵢ hᵢ·yᵢₜ) / qₜ ≤ H_max for all t.

Mass balance: qₜ = ∑ᵢ yᵢₜ for all t.

Initial and final inventory: zᵢ₀ = s₀ᵢ, zᵢ_{|T|} = sᴛᵢ.

The model determines the optimal monthly purchase and refining plan that maximizes profit while respecting capacity, storage, and product quality constraints.