Use Table 13.4 Inc needs to order a raw material to make a special polymer. The
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Question
Use Table 13.4 Inc needs to order a raw material to make a special polymer. The demand for the polymer is forecasted to be normally distributed with a mean of 300 gallons an d a standard deviation of 100 gallons. Goop sells the polymer for $23 per gallon. Goop purchases raw material for $9.5 per gallon and Goop must spend $4 per gallon to dispose of all unused raw material due to government regulations. (One gallon of raw material yields one gallon of polymer.) If demand is more than Goop can make, then only what has been made can be sold, and the remaining demand is lost. If a part of the question specifies whether to use Table 13.4, or to use Excel, then credit for a correct answer will depend on using the specified method. a. How many gallons should Goop purchase to maximize its expected profit? b. Suppose Goop purchases 125 gallons of raw material. What is the probability that it will run out of raw material? Use Table 13.4 (Round your answer to 4 decimal places.) Suppose Goop purchases 300 gallons of raw material. What is the expected sales (in gallons)? Use Table 13.4. (Round your answer to 2 decimal places.) Suppose Goop purchases 375 gallons of raw material. How much should it expect to spend on disposal costs (in $s)? Use Table 13.4. Round your answer to 2 decimal places.) d Suppose Goop wants to ensure that there is a 91% probability that it will be able to satisfy e the customer's entire demand. How many gallons of the raw material should it purchase? Use Table 13.4.Explanation / Answer
Mean demand, = 300 gallons
SD of demand, = 100 gallons
Selling price, s = 23
Cost price, c = 9.5
Cost of disposal = 4
a) Underage cost, Cu = s - c = 23-9.5 = 13.5
Overage cost, Co = c+d = 9.5+4 = 13.5
Optimal service level, F(z) = Cu/(Cu+Co) = 13.5/(13.5+13.5) = 0.5
Look in the table 13.4 for F(z) value as determined above, corresponding z value = 0
Therefore, Gallons of raw material Goop should purchase to maximize profits (Q) = +z = 300+0*100 = 300 gallons
b) Gallons purchased (Q) = 125
z value = (Q-)/ = (125-300)/100 = -1.75
Look in table 13.4 for corresponding value of F(z) = 0.0403 (z = -1.75 is not in the table, so take average of (F(z) for z = -1.7 and -1.8 )
Probability that it will run out of raw material = 1 - F(z) = 1 - 0.0403 = 0.9597
c) Purchased quantity (Q) = 300 gallons
z value = (Q-)/ = (300-300)/100 = 0
Using table 13.4, corresponding value of I(z) = 0.3989
Leftover inventory V = I(z) = 100*0.3989 = 39.89 gallons
Expected Sales = Q - V = 300 - 39.89 = 260.11 gallons
d) Purchased quantity (Q) = 375 gallons
z value = (Q-)/ = (375-300)/100 = 0.75
Using table 13.4, corresponding value of I(z) = 0.8816 (z = 0.75 is not in the table, so take average of (F(z) for z = 0.7 and 0.8 )
Leftover inventory V = I(z) = 100*0.8816 = 88.16 gallons
Expected spend on disposal costs = V*d = 88.16*4 = $ 352.62
e) For 91% in-stock probablity, F(z) = 0.9100
Using table 13.4, corresponding z value = 1.3 + (0.9100-0.9032)*(1.4-1.3)/(0.9192-0.9032) = 1.3425 (F(z) = 0.9100 is not in the table, so interpolated )
Gallons of raw material to purchase = + z = 300+1.3425*100 = 434.25 gallons
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