Let's answer each part step by step: --- ## **a. Develop a Linear Programming Model to Evaluate Clarksville** We use **Data Envelopment Analysis (DEA)**, a linear programming technique, to evaluate the efficiency of decision-making units (DMUs)—here, restaurants. ### **DEA Model for Clarksville** Let \( \lambda_i \) (weights) be the proportion of resources/output from each restaurant \( i \) (where \( i = 1, 2, 3, 4, 5 \) for the five restaurants). **Objective:** Minimize inputs for Clarksville to achieve at least its current output, compared to a composite (virtual) restaurant made from the best practices of the others. #### **Variables** - \( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 \): weights for each restaurant (including Clarksville itself). #### **Input Constraints** Construct a composite restaurant using the weighted sum of the inputs of all restaurants that produces at least as much output as Clarksville, using no more input than Clarksville. \[ \begin{align*} 96\lambda_1 + 110\lambda_2 + 100\lambda_3 + 125\lambda_4 + 120\lambda_5 &\leq 110 \theta \quad \text{(Hours)} \\ 16\lambda_1 + 22\lambda_2 + 18\lambda_3 + 25\lambda_4 + 24\lambda_5 &\leq 22 \theta \quad \text{(Staff)} \\ 850\lambda_1 + 1400\lambda_2 + 1200\lambda_3 + 1500\lambda_4 + 1600\lambda_5 &\leq 1400 \theta \quad \text{(Supplies)} \\ \end{align*} \] #### **Output Constraints** The virtual composite must produce at least as much output as Clarksville: \[ \begin{align*} 3800\lambda_1 + 4600\lambda_2 + 4400\lambda_3 + 6500\lambda_4 + 6000\lambda_5 &\geq 4600 \quad \text{(Profit)} \\ 25\lambda_1 + 32\lambda_2 + 35\lambda_3 + 30\lambda_4 + 28\lambda_5 &\geq 32 \quad \text{(Market Share)} \\ 8.0\lambda_1 + 8.5\lambda_2 + 8.0\lambda_3 + 10.0\lambda_4 + 9.0\lambda_5 &\geq 8.5 \quad \text{(Growth Rate)} \\ \end{align*} \] #### **Other Constraints** \[ \lambda_i \geq 0, \quad \sum_{i=1}^5 \lambda_i = 1 \] #### **Objective Function** Minimize \( \theta \) (proportion of Clarksville’s inputs needed). --- ## **b. Solve the Model** Let's try to see if any restaurant or combination can **dominate Clarksville** (i.e., produce at least as much output with less input): ### **Clarksville's Data:** - Inputs: 110 hrs, 22 staff, $1400 - Outputs: $4600, 32%, 8.5% ### **Compare with Others:** #### Example: New Albany - Inputs: 125 hrs, 25 staff, $1500 (all higher than Clarksville) - Outputs: $6500, 30%, 10% (better profit and growth, but lower market share) #### Example: Jeffersonville - Inputs: 100 hrs, 18 staff, $1200 (all lower) - Outputs: $4400, 35%, 8.0% - Profit is lower ($4400 < $4600), but market share and growth are not worse. #### Let's try a composite of Jeffersonville and St Matthews. Set up the system: - Let \( \lambda_3 \) (Jeffersonville) and \( \lambda_5 \) (St. Matthews) - Try to match Clarksville's outputs: - Profit: \( 4400\lambda_3 + 6000\lambda_5 \geq 4600 \) - Market Share: \( 35\lambda_3 + 28\lambda_5 \geq 32 \) - Growth: \( 8.0\lambda_3 + 9.0\lambda_5 \geq 8.5 \) - \( \lambda_3 + \lambda_5 = 1 \) Solve for \( \lambda_3, \lambda_5 \): - From growth: \( 8.0\lambda_3 + 9.0(1-\lambda_3) \geq 8.5 \implies 8.0\lambda_3 + 9.0 - 9.0\lambda_3 \geq 8.5 \implies -1.0\lambda_3 \geq -0.5 \implies \lambda_3 \leq 0.5 \) - From market share: \( 35\lambda_3 + 28(1-\lambda_3) \geq 32 \implies 35\lambda_3 + 28 - 28\lambda_3 \geq 32 \implies 7\lambda_3 \geq 4 \implies \lambda_3 \geq 0.571 \) But \( \lambda_3 \leq 0.5 \) and \( \lambda_3 \geq 0.571 \) have no solution. Try other combinations or check if any single restaurant is more efficient. --- **In short, Clarksville is not strictly dominated by any single restaurant, but could be compared to a composite of others. For a complete solution, solving the LP with all constraints would be needed (often done with software), but from above, Clarksville is not strictly inefficient.** --- ## **c. Is Clarksville Relatively Inefficient?** - Clarksville is **not clearly inefficient** compared to the others, as no restaurant achieves all of Clarksville's outputs with less input. - However, New Albany achieves much higher profit and growth with only slightly more input, hinting at possible inefficiency in Clarksville. --- ## **d. Composite Restaurant Output vs. Clarksville** From above, **no single restaurant** achieves all outputs with less input. You'd need to solve the LP (using software) to find the exact composite. But, **New Albany** comes closest in profit and growth, with only slightly more input. --- ## **e. Recommended Restaurants for Improvement** - **Clarksville** should look at **New Albany** (for profit and growth) and **Jeffersonville** (for efficiency in input use, especially staff and supplies). - Studying **how New Albany achieves higher output with similar resources** would provide insights. --- ### **Summary Table** | Restaurant | Inputs (hrs, staff, $) | Outputs ($, %, %) | Efficiency (Input/Output) | |----------------|-----------------------|-------------------|---------------------------| | Bardstown | 96, 16, 850 | 3800, 25, 8.0 | Lower profit, less input | | Clarksville | 110, 22, 1400 | 4600, 32, 8.5 | --- (Target) | | Jeffersonville | 100, 18, 1200 | 4400, 35, 8.0 | Lower input, lower profit | | New Albany | 125, 25, 1500 | 6500, 30, 10.0 | Higher output, more input | | St. Matthews | 120, 24, 1600 | 6000, 28, 9.0 | Similar trend | --- ## **Final Answers** ### **a)** See linear programming model above. ### **b)** Model suggests Clarksville is not strictly inefficient, but near the efficiency frontier. ### **c)** Not strictly inefficient, but should study New Albany. ### **d)** Composite restaurant would likely use a mix of New Albany/Jeffersonville; New Albany has more output for slightly more input. ### **e)** Study New Albany (high output) and Jeffersonville (efficient input use). --- **If you need a fully solved LP with exact weights, that's best done with Excel Solver or similar software. The above is the step-by-step logic you would use.**