# 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:

â€¢ 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.

â€¢ 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?

https://brainmass.com/business/operations-research/numericals-for-inventory-model-642422

#### Solution Preview

Please see the attached Excel file for answers. Answers are on different worksheets.

1. Walmart Inventory

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

Demand is normally distributed

Mean Weekly Demad= 200 jars

Standard Deviationfor weekly demand= 30 jars

Purchasing Cost= $4.00 per jar

Holding Cost= $1.00 per week

Ordering Cost= $25.00 per order

Lead Time= 2 weeks

Economic Order Quantity (EOQ) Model

Q* = âˆš 2DS/H

Q* = Economic Order Quantity

D= Weekly demand= 200

S= Ordering Cost= 25

H= Holding Cost per unit= $1.00

Q* = âˆš 2DS/H= 100 = âˆš (2x 200 x $25 / $1)=

Answer: Wal Mart should order 100 jars at a time

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

Reorder point=Average demand during lead time + Safety stock

Average demand during lead time= average demand per week x constant lead time in weeks

"Safety stock= z x ÏƒdLT

"

ÏƒdLT = standard deviation of demand during lead time

z = number of standard deviations required to meet the cycle-service level

Mean Weekly Demad= 200 jars

Standard Deviationfor weekly demand= 30 jars

Lead Time= 2 weeks

z for 95% service level (from table)= 1.645 (from the z tables)

Average demand during lead time= 400 =200x2

ÏƒdLT = Ïƒd âˆš L= 42.43 =30x âˆš2

"Safety stock= z x ÏƒdLT

" 69.80 =1.645x42.43

Reorder point= 469.80 =400+69.8

Or rounding up= 470 jars

Answer: Reorder point= 470 jars

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

Mean Weekly Demad= 200 jars

Standard Deviationfor weekly demand= 30 jars

Lead Time= 2 weeks

z for 99% service level (from table)= 2.326

Average demand during lead time= 400 =200x2

ÏƒdLT = Ïƒd âˆš L= 42.43 =30x âˆš2

"Safety stock= z x ÏƒdLT

" 98.69 =2.326x42.43

Reorder point= 498.69 =400+98.69

Or rounding up= 499 jars

Additional safety stock= 29 jars =499-470

Answer: Reorder point= 499 jars

Additional safety stock= 29 jars

2. Gullo Sunglasses

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

"Optimal order: balance the expected costs of ordering too much with the expected costs of ordering too little

"

Overage Cost= the incremental per unit cost for any item that cannot be sold

Underage Cost= the incremental per unit ...

#### Solution Summary

3 numerical questions on inventory, ordering excess vs less than demand inventory, and process capability index have been answered.