Work

Overview of GreenWeave's Fabric Production

  • Company: GreenWeave

  • Focus: Eco-friendly fabrics

  • Fabric types:

    • Hemp-Blend (H)

    • CottonSoft (C)

    • SilkFlex (S)

Components of Fabric Production

  1. Materials Used:

    • Hemp Fibers (hf)

    • Organic Cotton (oc)

    • Recycled Silk (rs)

  2. Sales Information:

    • Selling Price per Kilogram:

      • HempBlend: £25.00

      • CottonSoft: £22.00

      • SilkFlex: £30.00

Blending Constraints for Fabrics

  • HempBlend (H):

    • At least 60% Hemp Fibers

  • CottonSoft (C):

    • At most 20% Hemp Fibers

    • At least 60% Organic Cotton

  • SilkFlex (S):

    • At most 20% Hemp Fibers

    • At least 60% Recycled Silk

Supplier Constraints on Materials

  1. Hemp Fibers:

    • Maximum supply: 120 kg at a cost of £3.00/kg

  2. Organic Cotton:

    • Maximum supply: 100 kg at a cost of £2.50/kg

  3. Recycled Silk:

    • Maximum supply: 80 kg at a cost of £4.00/kg

Objective

  • Goal: Maximize profit from fabric production.

Linear Programming Formulation

Variables:

  • Let x<sub>H,hf</sub> = amount of Hemp Fibers used in HempBlend

  • Let x<sub>H,oc</sub> = amount of Organic Cotton used in HempBlend

  • Let x<sub>H,rs</sub> = amount of Recycled Silk used in HempBlend

  • Let x<sub>C,hf</sub> = amount of Hemp Fibers used in CottonSoft

  • Let x<sub>C,oc</sub> = amount of Organic Cotton used in CottonSoft

  • Let x<sub>C,rs</sub> = amount of Recycled Silk used in CottonSoft

  • Let x<sub>S,hf</sub> = amount of Hemp Fibers used in SilkFlex

  • Let x<sub>S,oc</sub> = amount of Organic Cotton used in SilkFlex

  • Let x<sub>S,rs</sub> = amount of Recycled Silk used in SilkFlex

Objective Function:

  • Maximize Profit (P): [ P = 25 * (x_{H,hf} + x_{H,oc} + x_{H,rs}) + 22 * (x_{C,hf} + x_{C,oc} + x_{C,rs}) + 30 * (x_{S,hf} + x_{S,oc} + x_{S,rs}) ]

Subject to Constraints:

  1. Hemp Fibers Usage Constraints: [ x_{H,hf} + x_{C,hf} + x_{S,hf} \leq 120 ]

  2. Organic Cotton Usage Constraints: [ x_{H,oc} + x_{C,oc} + x_{S,oc} \leq 100 ]

  3. Recycled Silk Usage Constraints: [ x_{H,rs} + x_{C,rs} + x_{S,rs} \leq 80 ]

  4. Mixing Constraints for Each Fabric:

    • HempBlend (H):[ x_{H,hf} \geq 0.6 * (x_{H,hf} + x_{H,oc} + x_{H,rs}) ]

    • CottonSoft (C): [ x_{C,hf} \leq 0.2 * (x_{C,hf} + x_{C,oc} + x_{C,rs}) ] [ x_{C,oc} \geq 0.6 * (x_{C,hf} + x_{C,oc} + x_{C,rs}) ]

    • SilkFlex (S):[ x_{S,hf} \leq 0.2 * (x_{S,hf} + x_{S,oc} + x_{S,rs}) ] [ x_{S,rs} \geq 0.6 * (x_{S,hf} + x_{S,oc} + x_{S,rs}) ]

Non-negativity Constraints:

  • All variables must be greater than or equal to zero: [ x_{H,hf}, x_{H,oc}, x_{H,rs}, x_{C,hf}, x_{C,oc}, x_{C,rs}, x_{S,hf}, x_{S,oc}, x_{S,rs} \geq 0 ]