1. **Problem Statement:** A manufacturer must decide between building a small or large factory with different costs and capacities. Demand levels are 10,000, 20,000, 50,000, and 100,000 pairs. Profit per pair is 10. We construct a payoff table and find the optimal decision using Laplace and Hurwicz criteria (40% optimism). 2. **Payoff Table Construction:** - Profit = (min(demand, capacity) * 10) - cost - Small factory cost = 150,000, capacity = 50,000 - Large factory cost = 300,000, capacity = 100,000 | Demand | Small Factory Payoff | Large Factory Payoff | |--------|---------------------|---------------------| | 10,000 | (10,000*10)-150,000 = -50,000 | (10,000*10)-300,000 = -200,000 | | 20,000 | (20,000*10)-150,000 = 50,000 | (20,000*10)-300,000 = -100,000 | | 50,000 | (50,000*10)-150,000 = 350,000 | (50,000*10)-300,000 = 200,000 | |100,000 | (50,000*10)-150,000 = 350,000 | (100,000*10)-300,000 = 700,000 | 3. **Laplace Criterion:** - Assumes equal probability for all demand levels. - Calculate average payoff for each factory: - Small: $\frac{-50,000 + 50,000 + 350,000 + 350,000}{4} = 175,000$ - Large: $\frac{-200,000 - 100,000 + 200,000 + 700,000}{4} = 150,000$ - Optimal decision: Build the small factory (higher average payoff). 4. **Hurwicz Criterion (40% optimism):** - Formula: $H = \alpha \times \text{max payoff} + (1-\alpha) \times \text{min payoff}$ where $\alpha=0.4$ - Small factory: max = 350,000, min = -50,000 - $H_s = 0.4 \times 350,000 + 0.6 \times (-50,000) = 140,000 - 30,000 = 110,000$ - Large factory: max = 700,000, min = -200,000 - $H_l = 0.4 \times 700,000 + 0.6 \times (-200,000) = 280,000 - 120,000 = 160,000$ - Optimal decision: Build the large factory (higher Hurwicz value). --- "slug":"factory decision","subject":"decision theory","desmos":{"latex":"y=\max(\min(x,50000)*10-150000,\min(x,100000)*10-300000)","features":{"intercepts":true,"extrema":true}},"q_count":5}