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1. Let x represent the number of units produced and sold, and let P represent the total profit. With aprofit of $100 per unit.a. What is the mathematical model for the total profit earned by producing and selling x units ?b. Furthermore, suppose the production capacity constraint is given by5x 40x 0What is the optimal solution for this production model?2. The W Company purchases a certain part of the engine from suppliers X, Y, and Z. Supplier X supplies50% of the parts with 3% defective rate. Supplier Y supplies 30% of the parts with 2% defective rate.Supplier Z supplies 20% of the supplies with 1% defective rate. When a defective part is found, whichsupplier is the most likely source?3. The average stock price for companies making up the Standard & Poor 500 was $30 per share and thestandard deviation was $8.20 in 2003. Suppose the stock prices are normally distributed, how high doesa stock price have to be to put a company in the top 1% of S&P500 ?4. Southland Corporationâs decision to produce a new line of products resulted in the need to constructeither a small factory or a large factory. For a small factory, the projected profit of $15 million in theevent of low demand, $20 million in the event of medium demand and $25 million in the event of highdemand. For a large factory, the projected profit of $5 million in the event of low demand, $20 million inthe event of medium demand and $50 million in the event of high demand. Furthermore, the probabilityof a low demand is 0.6, the probability of a medium demand is 0.3 and the probability of a high demandis 0.2. What is the optimal decision based on the expected value approach?5. In California, a lottery ticket costs $1. The jackpot prize is $5800000. Suppose that the chance ofwinning the jackpot is 1 in 1000000. Furthermore, suppose that a Californian assigns an indifferenceprobability of 0.000001 to the $0 payoff. Based on the expected utility approach, would this personpurchase a lottery ticket and why ?6. Red Army and Blue Army must decide whether to attack or defend their territories. The decisions aremade without the knowledge of the opposing armyâs decision in this zero-sum game. The Red Army willgain 3 territories if both armies decide to attack. The Red Army will gain 5 territories if it decides toattack and the Blue Army decides to defend. On the other hand, the Red Army will gain 4 territoriesif it decides to defend and the Blue Army decides to attack. The Red Army will gain nothing if botharmies decide to defend their territories. What is the optimal strategy for each army ?7. Consider the time series data.Month V alue1 212 143 194 135 196 227 16a. Construct a time series plot for the above set of data. What type of pattern exists in the data ?Page 1 of 2A final assessment for Math 2326b. Using 3-month moving average, what is the forecast for month 8 ?c. Using the exponential smoothing method with = 0.3, what is the forecast for month 8 ?d. Which of the above methods give a better forecast and why ?
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