1. **Problem 1 (Two marks):** Calculate the efficiency score of a Decision Making Unit (DMU) with inputs $x_1=4$, $x_2=3$ and outputs $y_1=8$, $y_2=6$ using the simple input-oriented DEA model. **Step 1:** State the problem: Find efficiency score $\theta$ minimizing inputs while producing at least the given outputs. **Step 2:** The input-oriented DEA efficiency is given by $$\theta = \frac{\text{weighted sum of inputs}}{\text{weighted sum of outputs}}$$ **Step 3:** For this simple case, efficiency $\theta = \frac{4+3}{8+6} = \frac{7}{14} = 0.5$. **Step 4:** Interpretation: The DMU is 50% efficient, meaning it could reduce inputs by half while maintaining outputs. 2. **Problem 2 (Six marks):** Given three DMUs with inputs and outputs: - DMU A: inputs $(2,3)$, outputs $(5,4)$ - DMU B: inputs $(3,2)$, outputs $(6,5)$ - DMU C: inputs $(4,4)$, outputs $(7,6)$ Calculate the efficiency score of DMU C using input-oriented CCR DEA model. **Step 1:** State the problem: Find $\theta$ minimizing inputs for DMU C while outputs are at least maintained. **Step 2:** The CCR model solves $$\min \theta$$ subject to $$\sum \lambda_j x_{ij} \leq \theta x_{iC}$$ and $$\sum \lambda_j y_{rj} \geq y_{rC}$$ for all inputs $i$ and outputs $r$, with $\lambda_j \geq 0$. **Step 3:** Set up constraints: $$2\lambda_A + 3\lambda_B + 4\lambda_C \leq \theta \times 4$$ $$3\lambda_A + 2\lambda_B + 4\lambda_C \leq \theta \times 4$$ $$5\lambda_A + 6\lambda_B + 7\lambda_C \geq 7$$ $$4\lambda_A + 5\lambda_B + 6\lambda_C \geq 6$$ **Step 4:** Solve the linear program (assuming $\lambda_C=0$ for efficiency calculation): Try $\lambda_A=1$, $\lambda_B=0$: Inputs: $2 \leq 4\theta$, $3 \leq 4\theta$ so $\theta \geq 0.75$ Outputs: $5 \geq 7$ (false), so increase $\lambda_B$. Try $\lambda_A=0.5$, $\lambda_B=0.5$: Inputs: $2(0.5)+3(0.5)=2.5 \leq 4\theta$, $3(0.5)+2(0.5)=2.5 \leq 4\theta$ so $\theta \geq 0.625$ Outputs: $5(0.5)+6(0.5)=5.5 \geq 7$ (false), increase $\lambda_B$. Try $\lambda_B=1$: Inputs: $3 \leq 4\theta$, $2 \leq 4\theta$ so $\theta \geq 0.75$ Outputs: $6 \geq 7$ (false). **Step 5:** Since no combination satisfies outputs, DMU C is efficient with $\theta=1$. **Step 6:** Final answer: Efficiency score of DMU C is 1 (fully efficient).