# Numericals for Inventory Model

1. Walmart Inventory
As a manager at Wal-Mart, you have just realized that the store frequently runs out of Nutella spread jars. You are suspecting that there might be a problem with the replenishment policy.

Thus, you have decided to gather more data.

The demand for Nutella turns out to be normally distributed. The mean demand for one week is 200 jars and the standard deviation for the weekly demand is 30 jars. Wal-Mart's purchasing cost for Nutella is \$4 per jar. The cost of holding one Nutella jar in the inventory is \$1 per week. In addition, the store incurs a cost of \$25 for placing an order and it takes 2 weeks to receive the order.

Wal-Mart is using continuous time (Q,ROP) policy to replenish its Nutella jar inventory. Based on this, please answer the following questions. Please assume that 1 year = 52 weeks. (Please note that z-table is provided on page 2.)

a. How many Nutella jars should Wal-Mart order at a time?

b. What should be the reorder point if Wal-Mart wants to provide a 95% service level?

c. How much additional safety stock is needed if the service level is increased from 95% to 99%?

Table of z-values
Probability z-value Probability z-value
0.99 2.326 0.89 1.227
0.98 2.054 0.88 1.175
0.97 1.881 0.87 1.126
0.96 1.751 0.86 1.080
0.95 1.645 0.85 1.036
0.94 1.555 0.84 0.994
0.93 1.476 0.83 0.954
0.92 1.405 0.82 0.915
0.91 1.341 0.81 0.878
0.90 1.282 0.80 0.842

2. Gullo Sunglasses
Geoff Gullo owns a small firm that manufactures "Gullo Sunglasses." He has the opportunity to sell a particular seasonal model to Land's End. Geoff offers Land's End two purchasing options:
• Option 1: Geoff offers to set his price at \$65 and agrees to credit Land's End \$40 for each unit Land's End returns to Geoff at the end of the season (because those units did not sell). Since styles change each year, there is essentially no value in the returned merchandise.

• Option 2: Geoff offers a price of \$55 for each unit, but returns are no longer accepted. In this case, Land's End throws out unsold units at the end of the season.

The season's demand for this model will be normally distributed with mean of 200 and standard deviation of 125. Land's End will sell those sunglasses for \$100 each. Geoff's production cost is \$25

(a) How many pairs of sunglasses would Land's End buy under option 1? Option 2?

(b) To help Land's End choose which option to take, you have run a Crystal Ball simulation (see Table 1 on next page). For a given quantity of sunglasses purchased, you simulated demand 1,000 times to estimate how many sunglasses would be sold and how many sunglasses would be left over. Based on this information, which option will Land's End choose?

Table 1: Summary Output of Crystal Ball Simulation
Sunglasses
Bought Sold Leftover
160 131 29
165 134 31
170 137 33
175 140 35
180 142 38
185 145 40
190 148 42
195 151 44
200 153 47
205 156 49
210 158 52
215 160 55
220 162 58
225 165 60
230 167 63
235 169 66
240 171 69
245 172 73
250 174 76
255 176 79
260 177 83
265 179 86
270 180 90
275 182 93
280 183 97
285 184 101
290 186 104
295 187 108
300 188 112

3. Process Capability Index
Krombach Auto Corporation, a German manufacturer of cars, investigates their steering wheel attachment process. A steering wheel needs some "Spiel", a small range where moving the steering wheel does not turn the tires. This increase driving safety, as not every small movement by the often-nervous driver is picked up by the car. However, if the Spiel is too large, drivers will feel uncomfortable and have the impression that the car does not react to their action. Krombach estimates that a Spiel of less than 1 degree is too little and causes too much nervousness. They
also estimate that a Spiel of more than 3 degrees is too much and causes concern with the drivers. The steering wheel attachment process results in a Spiel with a mean of 1.7 degrees and a standard deviation of 0.4 degrees. Is the process capable according to the traditional Cpk definition of process capability?