1. **Problem Statement:** A manufacturer produces two products A and B using limited labour and raw materials. 2. **Given Data:** - Product A requires 2 hours labour and 3 units raw materials per unit. - Product B requires 4 hours labour and 4 units raw materials per unit. - Available daily: 28 hours labour, 32 units raw materials. - Profit per unit: A = 8, B = 6. - Goals: a) Profit \( \geq 44 \) b) Units of A = 2 \times units of B c) Labour time fully utilized (28 hours). 3. **Decision Variables:** Let \( x_A \) = units of product A produced Let \( x_B \) = units of product B produced 4. **Goal Programming Model Formulation:** We introduce deviation variables to measure under- or over-achievement of goals: - \( d_1^- \), \( d_1^+ \): under- and over-achievement of profit goal - \( d_2^- \), \( d_2^+ \): under- and over-achievement of production ratio goal - \( d_3^- \), \( d_3^+ \): under- and over-utilization of labour time 5. **Profit Goal:** Profit = \( 8x_A + 6x_B \) Goal: \( 8x_A + 6x_B + d_1^- - d_1^+ = 44 \) 6. **Production Ratio Goal:** Goal: \( x_A - 2x_B + d_2^- - d_2^+ = 0 \) 7. **Labour Time Utilization Goal:** Labour used = \( 2x_A + 4x_B \) Goal: \( 2x_A + 4x_B + d_3^- - d_3^+ = 28 \) 8. **Resource Constraints:** Raw materials used: \( 3x_A + 4x_B \) must be \( \leq 32 \) 9. **Non-negativity:** \( x_A, x_B, d_i^-, d_i^+ \geq 0 \) for all deviation variables. **Summary of the Goal Programming Model:** $$ \begin{cases} 8x_A + 6x_B + d_1^- - d_1^+ = 44 \\ x_A - 2x_B + d_2^- - d_2^+ = 0 \\ 2x_A + 4x_B + d_3^- - d_3^+ = 28 \\ 3x_A + 4x_B \leq 32 \\ x_A, x_B, d_i^-, d_i^+ \geq 0 \end{cases} $$ This model allows the manufacturer to minimize deviations from the goals while respecting resource limits.