Purchasing road salt for towns in the Northeast is a challenging task. The town of Homer, New York has calculated a forecast of their annual salt needs using historical data. The forecast is summarized in the table below (Q is the quantity needed):
For example, there is a 60.6% chance they will need 50,000 tons or fewer, there is a 3.3% chance they will need exactly 100,000 tons and there is a very small chance they will need more than 200,000 tons. Suppose Homer wants to minimize the amount of inventory it purchases while at the same time having no more than a 6% change of running out of salt (which would force it to purchase salt on the spot market for a premium).
a. At the start of the season, how much salt (in tons) should Homer have available in its storage sheds? Assume salt must be purchased in increments of 10,000.
|
Q (ton)
(a)
|
Distn Fn
|
Density Fn
(b)
|
c=a*b
|
|
0
|
0.0%
|
0.0%
|
0
|
|
10000
|
9.7%
|
9.7%
|
970
|
|
20000
|
23.9%
|
14.2%
|
2840
|
|
30000
|
37.8%
|
13.9%
|
4170
|
|
40000
|
50.2%
|
12.4%
|
4960
|
|
50000
|
60.6%
|
10.4%
|
5200
|
|
60000
|
69.1%
|
8.5%
|
5100
|
|
70000
|
76.0%
|
6.9%
|
4830
|
|
80000
|
81.4%
|
5.4%
|
4320
|
|
90000
|
85.7%
|
4.3%
|
3870
|
|
100000
|
89.0%
|
3.3%
|
3300
|
|
110000
|
91.6%
|
2.6%
|
2860
|
|
120000
|
93.6%
|
2.0%
|
2400
|
|
130000
|
95.1%
|
1.5%
|
1950
|
|
140000
|
96.3%
|
1.2%
|
1680
|
|
150000
|
97.2%
|
0.9%
|
1350
|
|
160000
|
97.9%
|
0.7%
|
1120
|
|
170000
|
98.4%
|
0.5%
|
850
|
|
180000
|
98.8%
|
0.4%
|
720
|
|
190000
|
99.1%
|
0.3%
|
570
|
|
200000
|
99.3%
|
0.2%
|
400
|
|
|
|
Mean
|
53460
|
|
|
|
Std Dev
|
1749
|
Stockout Probailbiy = 6% = 0.06
So Instock Prob = 1-0.06 = 0.94
Corresponding z-stat is NORM.S.INV(0.94) = 1.5548
Z = (S-Mean)/Std Dev
So S = Mean + z*Std Dev = 53460 + 1.5548*1749 = 56179
As Order is in multiple of 10,000, So Order Qty will be 60,000
Order Quantity (tons) 60,000
b. Now suppose Homer has been offered the following deal from American Salt Mine (ASM). ASM will sell Homer salt options for $30 per option with an exercise price of $40 for each option Homer purchases in advance of the season, Homer can "exercise" an option during the season to receive one ton of salt during the season. For example, if Homer purchases 100,000 options before the season starts, then it pays ASM $30 x 100,000 = $3,000,000 for those options. As Homer needs salt during the season, ASM will deliver up to 100,000 tons for a price of $40 per ton. Options are good only for this season - any unexercised options at the end of the season have no value. Finally, if Homer exercises all of its options and still needs more salt, then it will have to purchase salt in the spot market, for an estimated $80 per ton. Given this deal, how many options should Homer purchase from ASM? Assume options must be purchased in increments of 10,000 tons.
Co = Overage Cost = Option Cost = 30
Cu = Underage cost = Mkt Price - Option excerise Cost = 80-40 = 40
So Critical Ratio F(Q) = Cu/(Co+Cu) = 40/(30+40) = 0.5714
Corresponding z-stat is NORM.S.INV(0.5714) = 0.1799
So Optimal Order Qty Q = Mean + Std Dev*z = 53460 + 1749*0.1799 = 53775 ton
So Homer should buy 60,000 Options