1. Let x represent the number of units produced and sold, and let P represent the total profit. With a
profit 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 by
5x 40
x 0
What 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 supplies
50% 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, which
supplier is the most likely source?
3. The average stock price for companies making up the Standard & Poor 500 was $30 per share and the
standard deviation was $8.20 in 2003. Suppose the stock prices are normally distributed, how high does
a stock price have to be to put a company in the top 1% of S&P500 ?
4. Southland Corporation's decision to produce a new line of products resulted in the need to construct
either a small factory or a large factory. For a small factory, the projected profit of $15 million in the
event of low demand, $20 million in the event of medium demand and $25 million in the event of high
demand. For a large factory, the projected profit of $5 million in the event of low demand, $20 million in
the event of medium demand and $50 million in the event of high demand. Furthermore, the probability
of a low demand is 0.6, the probability of a medium demand is 0.3 and the probability of a high demand
is 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 of
winning the jackpot is 1 in 1000000. Furthermore, suppose that a Californian assigns an indifference
probability of 0.000001 to the $0 payoff. Based on the expected utility approach, would this person
purchase a lottery ticket and why ?
6. Red Army and Blue Army must decide whether to attack or defend their territories. The decisions are
made without the knowledge of the opposing army's decision in this zero-sum game. The Red Army will
gain 3 territories if both armies decide to attack. The Red Army will gain 5 territories if it decides to
attack and the Blue Army decides to defend. On the other hand, the Red Army will gain 4 territories
if it decides to defend and the Blue Army decides to attack. The Red Army will gain nothing if both
armies decide to defend their territories. What is the optimal strategy for each army ?
7. Consider the time series data.
Month Value
1 21
2 14
3 19
4 13
5 19
6 22
7 16
a. Construct a time series plot for the above set of data. What type of pattern exists in the data ?
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A final assessment for Math 2326
b. 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 ?