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Section 2
Discuss the characteristics of optimisation problems
What is optimisation?
Finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual of a business.
Explore the use of linear programming (LP) in solving optimisation problems
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2Febcbe79c-91a8-4ed0-ab0f-fe197e1ff958%2FUntitled.png%3Fid%3Dba44fcea-41a9-4657-9cf4-fc1c4664ca0f%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DwaVmMXU1jdtZq-HBMTGDHfBEQIcKzTnZZPaMDU_gwdA?table=block&id=ba44fcea-41a9-4657-9cf4-fc1c4664ca0f&cache=v2)
- Understand the problem.
- Define the decision variables.
X_1 =number of Aqua-Spas to produce
X_2 =number of Hydro-Luxes to produce
- Define the objective function
MAX: 350X 1 + 300X 2
- Define the constraints
- For this step, there are types of constraints in other cases, namely Demand Constraints, Resource Constraints, and Non-negativity conditions (which the next step).
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2F29d7880c-c0eb-49c6-8aab-64c4c47ae246%2FUntitled.png%3Fid%3D1df5058d-5957-419b-aaa1-b189364954d9%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DhqFAiYB_Vu8zFcY20MlTkXCLKP9-SBlvnGDycoWlNCI?table=block&id=1df5058d-5957-419b-aaa1-b189364954d9&cache=v2)
- Identify any upper or lower bounds on the decision variables.
X1 >= 0 && X2 >= 0
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2Fabb79e9a-2b81-4817-aa0d-7692e7effd19%2FUntitled.png%3Fid%3D0e09c5c4-01b1-4c63-9821-2312da1490bb%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DhOqa87eKDrJ2ee-d1M8PCAqgqdmUZDz2h6ACgT30i2E?table=block&id=0e09c5c4-01b1-4c63-9821-2312da1490bb&cache=v2)
Solving LP problems using graphical approach
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2F95e7f9c8-231d-4cdc-b5c5-4b9bd37813af%2FUntitled.png%3Fid%3D5fd13656-8917-4d4b-9f3f-4370d9496260%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DEktZpFfytaPFzG49MkfJlQbkRiBXJYuuecvV5KpLXSY?table=block&id=5fd13656-8917-4d4b-9f3f-4370d9496260&cache=v2)
![In Summary](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2F7de67790-9660-4918-9ca0-aa83f89b3d6f%2FUntitled.png%3Fid%3D9a3af409-ffd8-4557-b4d9-a30ae8894af1%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DUiie2AN7fQUkhNSkSF0CbIUrxyf2srPj7gSH_UJLrKs?table=block&id=9a3af409-ffd8-4557-b4d9-a30ae8894af1&cache=v2)
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2Fe5fd62c9-6a34-4fa6-a300-ddc80c8ffc56%2FUntitled.png%3Fid%3D1eff239c-ecfa-4f30-b77b-849d4189877e%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DFsU07lHlvsdnM6eqSvI3zxFEL7jWnQV3_zCtq09A6a4?table=block&id=1eff239c-ecfa-4f30-b77b-849d4189877e&cache=v2)
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2F884fea56-896d-40fb-8130-bab7df3410fa%2FUntitled.png%3Fid%3D73cb1189-d343-4201-bc81-3e74b6e218a3%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DPCGGb4ve9Gf7kQmDkFEdQazUuTdB6om77nN_4Kj_8Dc?table=block&id=73cb1189-d343-4201-bc81-3e74b6e218a3&cache=v2)
Special condition
Case 1: Alternate Optimal Solutions
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2Fe2cadb8f-b336-4d34-87a3-3563528a70f1%2FUntitled.png%3Fid%3D11c111fc-2534-44dd-8df0-529f1455aab9%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DhfGpgLG5Uy_heQberkRB8Wveb6Qw0uQcBRfGR1LnmcY?table=block&id=11c111fc-2534-44dd-8df0-529f1455aab9&cache=v2)
Case 2: Redundant Constraint
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2Fcfe36483-c771-4824-a67f-8e6daa8eb742%2FUntitled.png%3Fid%3D4d55596e-75d2-4ba8-8df6-713c188d4dc5%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DgnjFCaImzKwQeB9nBijGtBTzGfYcQ6l962_sRrZkFBg?table=block&id=4d55596e-75d2-4ba8-8df6-713c188d4dc5&cache=v2)
Case 3: Unbounded Solution
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2Fb9df9256-815f-484e-9af1-254b903a5ddb%2FUntitled.png%3Fid%3Dbd6a1133-7753-4e4c-a651-b34f2b7bb9cc%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DBvSyzCYI6DDvDIyVdBgxfzaFDSgexYPzp8xnHVJjyoo?table=block&id=bd6a1133-7753-4e4c-a651-b34f2b7bb9cc&cache=v2)
Case 4: Infeasibility
![notion image](https://www.notion.so/image/https%3A%2F%2Ffile.notion.so%2Ff%2Ff%2Fe1c5ba0e-562a-49b0-8505-ce5e38fc061b%2F972372be-95ab-4373-b9f8-f64f129d431b%2FUntitled.png%3Fid%3D94ddd2ac-8bd3-4512-a156-f8e37c8218eb%26table%3Dblock%26spaceId%3De1c5ba0e-562a-49b0-8505-ce5e38fc061b%26expirationTimestamp%3D1721066400000%26signature%3DpRP6IWOquMFZsDiWjQGsBqU3dMq2KLuvv1rm29GCYig?table=block&id=94ddd2ac-8bd3-4512-a156-f8e37c8218eb&cache=v2)
- Author:Jason Siu
- URL:https://jason-siu.com/article%2Fa5710386-b967-4116-bf5c-d219298c4dbf
- Copyright:All articles in this blog, except for special statements, adopt BY-NC-SA agreement. Please indicate the source!
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