Reorder Point Calculation for Inventory Management with Normal Distribution

Math Problem Statement

The table below shows the total area under the normal curve for a point that is Z standard deviations to the right of the mean. Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5754 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7258 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7518 0.7549 0.7 0.7580 0.7612 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7996 0.8023 0.8051 0.8079 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9430 0.9441 1.6 0.9452 0.9463 0.9474 0.9485 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9700 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9762 0.9767 2.0 0.9773 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9865 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9980 0.9980 0.9981 2.9 0.9981 0.9982 0.9983 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 Sam's Pet Hotel operates 52 weeks per year, 7 days per week, and uses a continuous review inventory system. It purchases kitty litter for $11.50 per bag. The following information is available about these bags. Refer to the standard normal tableLOADING... for z-values. followsDemand = 96 bags/week followsOrder cost = $57/order followsAnnual holding cost = 29 percent of cost followsDesired cycle-service levelequals92 percent followsLead time = 2 week(s) (14 working days) followsStandard deviation of weekly demand = 20 bags followsCurrent on-hand inventory is 300 bags, with no open orders or backorders. Part 2 a. What is the EOQ? Sam's optimal order quantity is 413 bags. (Enter your response rounded to the nearest whole number.) Part 3 What would be the average time between orders (in weeks)? The average time between orders is 4.3 weeks. (Enter your response rounded to one decimal place.) Part 4 b. What should R be? The reorder point is enter your response here bags. (Enter your response rounded to the nearest whole number.)

Solution

To calculate the reorder point (R) for Sam's Pet Hotel's inventory system, we use the following formula:

$R = (d \cdot L) + (z \cdot \sigma_L)$

Where:

$d$ : Average weekly demand = 96 bags/week
$L$ : Lead time (in weeks) = 2 weeks
$z$ : z-value corresponding to the desired cycle service level of 92%
$\sigma_L$ : Standard deviation of demand during lead time

Step 1: Calculate $\sigma_L$ (Lead time demand variability)

$\sigma_L = \sigma_d \cdot \sqrt{L}$

$\sigma_d$ : Standard deviation of weekly demand = 20 bags
$L = 2$ weeks

$\sigma_L = 20 \cdot \sqrt{2} = 20 \cdot 1.414 = 28.28 \, \text{bags}$

Step 2: Find $z$ -value for 92% service level

From the provided standard normal table, the cumulative area closest to 0.92 is 0.9192, which corresponds to $z = 1.41$ .

Step 3: Calculate $R$

$R = (d \cdot L) + (z \cdot \sigma_L)$ Substitute the values:

$d = 96$
$L = 2$
$z = 1.41$
$\sigma_L = 28.28$

$R = (96 \cdot 2) + (1.41 \cdot 28.28)$ $R = 192 + 39.88 = 231.88$

Step 4: Round $R$ to the nearest whole number

$R = 232 \, \text{bags}$

Final Answer:

The reorder point ( $R$ ) is 232 bags.

Do you have any questions or want detailed clarification? Here are 5 related questions you might consider:

How does lead time affect the reorder point calculation?
What happens to the reorder point if the cycle-service level is increased?
Can you explain the role of standard deviation in inventory management?
How is the EOQ related to the reorder point?
What would the impact of a higher weekly demand standard deviation be?

Tip: Always keep track of your lead time variability to adjust safety stock and ensure you meet service level goals!

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Math Problem Analysis

Mathematical Concepts

Inventory Management
Normal Distribution
Standard Deviation
Cycle Service Level
Reorder Point

Formulas

R = (d × L) + (z × σ_L)
σ_L = σ_d × sqrt(L)

Theorems

Properties of the Normal Distribution
Z-Score Table Application

Suitable Grade Level

Undergraduate Business or Operations Management

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