Let's break down the problem and **formulate a goal programming model** as requested in part (a). --- ## **Step 1: Define Decision Variables** Let: - \( x_1 \) = Number of units of part 1 to produce per year - \( x_2 \) = Number of units of part 2 to produce per year - \( x_3 \) = Number of units of part 3 to produce per year - \( x_4 \) = Number of units of part 4 to produce per year --- ## **Step 2: List the Objectives (Goals) and Constraints** ### **Resources and Limits** - **Stamping hours:** \(0.06x_1 + 0.17x_2 + 0.10x_3 + 0.14x_4 \leq 700\) - **Assembly hours:** \(0.18x_1 + 0.20x_2 + 0.08x_3 + 0.14x_4 \leq 700\) - **Finishing hours:** \(0.07x_1 + 0.20x_2 + 0.08x_3 + 0.12x_4 \leq 800\) - **Packaging hours:** \(0.09x_1 + 0.12x_2 + 0.07x_3 + 0.15x_4 \leq 600\) - **Sheet metal:** \(2.6x_1 + 1.4x_2 + 2.5x_3 + 3.2x_4 \leq 12 \times 15,000 = 180,000\) (since the demand and production are annual) ### **Demand (Goal 2):** - \(x_1 \leq 2,600\) - \(x_2 \leq 1,800\) - \(x_3 \leq 4,100\) - \(x_4 \leq 1,200\) ### **Profit (Goal 3):** - \(90x_1 + 100x_2 + 80x_3 + 120x_4 \geq 700,000\) --- ## **Step 3: Introduce Deviational Variables** For each goal, introduce deviational variables: - **For each resource constraint:** Let \(d_{i}^{-}\) and \(d_{i}^{+}\) be under- and over-utilization of resource \(i\) (where \(+\) means overtime, which we want to avoid). - **For demand:** Let \(d_{j}^{-}\) and \(d_{j}^{+}\) be under- and over-production for part \(j\). - **For profit:** Let \(d_P^{-}\) (underachievement of profit) and \(d_P^{+}\) (overachievement of profit). --- ## **Step 4: Write the Goal Constraints** ### **Resource Constraints (avoiding overtime):** \[ \begin{align*} 0.06x_1 + 0.17x_2 + 0.10x_3 + 0.14x_4 + d_1^{-} - d_1^{+} &= 700 \\ 0.18x_1 + 0.20x_2 + 0.08x_3 + 0.14x_4 + d_2^{-} - d_2^{+} &= 700 \\ 0.07x_1 + 0.20x_2 + 0.08x_3 + 0.12x_4 + d_3^{-} - d_3^{+} &= 800 \\ 0.09x_1 + 0.12x_2 + 0.07x_3 + 0.15x_4 + d_4^{-} - d_4^{+} &= 600 \\ \end{align*} \] - \(d_i^{+}\) represents overtime (to be minimized). ### **Demand Constraints:** \[ \begin{align*} x_1 + d_{d1}^{-} - d_{d1}^{+} &= 2,600 \\ x_2 + d_{d2}^{-} - d_{d2}^{+} &= 1,800 \\ x_3 + d_{d3}^{-} - d_{d3}^{+} &= 4,100 \\ x_4 + d_{d4}^{-} - d_{d4}^{+} &= 1,200 \\ \end{align*} \] - \(d_{dj}^{-}\): underproduction (to be minimized). ### **Profit Constraint:** \[ 90x_1 + 100x_2 + 80x_3 + 120x_4 + d_P^{-} - d_P^{+} = 700,000 \] - \(d_P^{-}\): underachievement of profit (to be minimized). ### **Material Constraint:** \[ 2.6x_1 + 1.4x_2 + 2.5x_3 + 3.2x_4 + d_M^{-} - d_M^{+} = 180,000 \] - \(d_M^{+}\): overuse (to be minimized). --- ## **Step 5: Set Up the Goal Programming Objective Function** Prioritize the goals as specified: - \(P_1\): Avoid overtime (\(\sum d_i^{+}\)) - \(P_2\): Meet demand (\(\sum d_{dj}^{-}\)) - \(P_3\): Achieve profit (\(d_P^{-}\)) - \(P_4\): Avoid overuse of material (\(d_M^{+}\)) The lexicographic (pre-emptive) goal programming objective function is: \[ \text{Minimize: } P_1 (\sum_{i=1}^4 d_i^{+}) + P_2 (\sum_{j=1}^4 d_{dj}^{-}) + P_3 (d_P^{-}) + P_4 (d_M^{+}) \] where \(P_1 \gg P_2 \gg P_3 \gg P_4\). --- ## **Step 6: Non-negativity Constraints** All variables and deviational variables must be non-negative. --- ## **Final Formulation** ### **Decision Variables:** - \(x_1, x_2, x_3, x_4 \geq 0\) ### **Deviational Variables:** - \(d_i^{-}, d_i^{+}, d_{dj}^{-}, d_{dj}^{+}, d_P^{-}, d_P^{+}, d_M^{-}, d_M^{+} \geq 0\) ### **Objective:** \[ \text{Minimize: } P_1 (\sum_{i=1}^4 d_i^{+}) + P_2 (\sum_{j=1}^4 d_{dj}^{-}) + P_3 (d_P^{-}) + P_4 (d_M^{+}) \] ### **Subject to:** \[ \begin{align*} 0.06x_1 + 0.17x_2 + 0.10x_3 + 0.14x_4 + d_1^{-} - d_1^{+} &= 700 \\ 0.18x_1 + 0.20x_2 + 0.08x_3 + 0.14x_4 + d_2^{-} - d_2^{+} &= 700 \\ 0.07x_1 + 0.20x_2 + 0.08x_3 + 0.12x_4 + d_3^{-} - d_3^{+} &= 800 \\ 0.09x_1 + 0.12x_2 + 0.07x_3 + 0.15x_4 + d_4^{-} - d_4^{+} &= 600 \\ x_1 + d_{d1}^{-} - d_{d1}^{+} &= 2,600 \\ x_2 + d_{d2}^{-} - d_{d2}^{+} &= 1,800 \\ x_3 + d_{d3}^{-} - d_{d3}^{+} &= 4,100 \\ x_4 + d_{d4}^{-} - d_{d4}^{+} &= 1,200 \\ 90x_1 + 100x_2 + 80x_3 + 120x_4 + d_P^{-} - d_P^{+} &= 700,000 \\ 2.6x_1 + 1.4x_2 + 2.5x_3 + 3.2x_4 + d_M^{-} - d_M^{+} &= 180,000 \\ x_1, x_2, x_3, x_4, d_i^{-}, d_i^{+}, d_{dj}^{-}, d_{dj}^{+}, d_P^{-}, d_P^{+}, d_M^{-}, d_M^{+} &\geq 0 \end{align*} \] --- ## **(a) This is the goal programming model as requested.** --- ### **(b) & (c)** - For (b), you would solve this using a linear programming solver. - For (c), if the first two priorities were reversed, you would minimize unmet demand before overtime, changing the order of \(P_1\) and \(P_2\) in the objective function. --- **If you need the solution for (b) or (c) or a numerical solution, let me know!**