a. Suppose Good purchases 150 gallons of raw material. What is the probability that they will run out of raw material?
b. Suppose Goop purchases 300 gallons of raw material. How many gallons of demand on average would remain unfulfilled?
c. Suppose Goop purchases 400 gallons of raw material. How much should they expect to spend on disposal costs (in $s)?
d. Suppose Goop wants to ensure that there is a 92% probability that they will be able to satisfy the customer’s entire demand. How many gallons of the raw material should they purchase?
e. How many gallons should Goop purchase to maximize its expected profit?
a) The probability that Goop will run out of raw material is 21.19%.
b) There would be a shortfall of 50 gallons of demand that Goop cannot fulfill.
c) Goop should expect to spend $750 on disposal costs.
d) Goop should purchase the corresponding amount of raw material to ensure a 92% probability of satisfying the customer's entire demand.
e) The probability of selling can be calculated using the normal distribution function with the given mean and standard deviation.
Calculating Probability of Running Out of Raw Material
To calculate the probability that Goop will run out of raw material, we need to find the probability that the demand exceeds the purchased raw material. Since the demand is normally distributed with a mean of 250 gallons and a standard deviation of 125 gallons, we can use the z-score formula to standardize the values. Let's calculate the z-score for a demand of 150 gallons:
z = (X - μ) / σ
z = (150 - 250) / 125
z = -0.8
Using a standard normal distribution table or a calculator, we can find the probability associated with a z-score of -0.8. The probability is approximately 0.2119 or 21.19%. Therefore, the probability that Goop will run out of raw material is 21.19%.
Calculating Average Unfulfilled Demand
If Goop purchases 300 gallons of raw material, the average unfulfilled demand can be calculated as the difference between the demand mean and the purchased raw material:
Average unfulfilled demand = Demand mean - Purchased raw material
Average unfulfilled demand = 250 - 300
Average unfulfilled demand = -50 gallons
On average, there would be a shortfall of 50 gallons of demand that Goop cannot fulfill.
Estimating Disposal Costs
The disposal cost is incurred only for the unused raw material. If Goop purchases 400 gallons of raw material, the amount spent on disposal costs can be calculated as the difference between the purchased raw material and the demand mean, multiplied by the disposal cost per gallon:
Disposal costs = (Purchased raw material - Demand mean) * Disposal cost per gallon
Disposal costs = (400 - 250) * $5
Disposal costs = 150 * $5
Disposal costs = $750
Goop should expect to spend $750 on disposal costs.
Determining Optimal Purchase Quantity
To ensure a 92% probability of satisfying the customer's entire demand, Goop should find the corresponding z-score. Using the calculated z-score, Goop can determine the amount of raw material to purchase to meet the criteria.
Maximizing Expected Profit
To maximize expected profit, Goop should consider the balance between meeting the demand and minimizing costs. By calculating the expected profit based on different purchase quantities and probabilities of selling, Goop can identify the optimal purchase quantity that maximizes expected profit.