# MATHEMATICS

Problem No. 1

A machine produces a product and the operator of the machine would like to develop an SPC chart to control the proportion of non-conforming (defective) units produced. She/he takes twenty samples of 250 units each and records the number of defective units found in each sample in the following table:

Sample
No.
Number of
Defective Units
Sample
No.
Number of
Defective Units
1 8 11 4
2 5 12 3
3 3 13 5
4 9 14 6
5 4 15 2
6 5 16 5
7 8 17 0
8 5 18 3
9 3 19 4
10 6 20 2

A. [2 points] What type of control chart (X-bar, R, p, or c) is appropriate for this process? Why?  (No explanation, no credit)

B. [4 points] Calculate the control limits for a 2-sigma control chart for this process.

C. [4 points] Given the sample data above, is the process in or out of control?  EXPLAIN (be specific as to why). No explanation, no credit! You must draw a control chart (Use Excel for this purpose and copy the graph into your Word document).

Problem No. 2

A chocolate manufacturer would like to develop a process control chart to control the weight of chocolate bars it produces.  Historically, the process has had a standard deviation of 0.25.  The operator has taken 39 samples of size 7 each and calculated the following sample means:

Sample
#
Sample
Mean
Sample
No.
Sample
Mean
Sample
No.
Sample
Mean
1 3.86 14 3.81 27 3.81
2 3.90 15 3.83 28 3.86
3 3.83 16 3.86 29 3.98
4 3.81 17 3.82 30 3.96
5 3.84 18 3.86 31 3.88
6 3.83 19 3.84 32 3.76
7 3.87 20 3.87 33 3.83
8 3.88 21 3.84 34 3.77
9 3.84 22 3.82 35 3.86
10 3.80 23 3.89 36 3.80
11 3.88 24 3.86 37 3.84
12 3.86 25 3.88 38 3.79
13 3.88 26 3.90 39 3.85

A. [3 points] Calculate the control limits for a 2-sigma x-bar control chart for this process.
B. [3 points] Assuming the R-chart for the process is in control, is this process in control? Why or Why not?  EXPLAIN (be specific as to why). No explanation, no credit.
C. [4 points] The specifications for the product are as follows:  3.8 ± 0.4 oz.  Calculate the Capability ratio and index.  Interpret them (be specific).

Problem No. 3
An appliance store carries a certain brand of TV which has he following characteristics:

Average daily demand 2 units
Ordering cost  \$25 per order
Carrying Cost 35%  of unit cost per year
Unit cost \$400 per unit
Standard deviation of daily demand 0.8 unit
Standard deviation of lead time 0.6 days

The firm currently orders the product 85 units at a time and operates 250 days a year.

A. [4 points] With the current lot-size policy, what is the annual holding and ordering costs?

B. [3 points] With the current lot size, what is the average time (in days) between orders?

C. [3 points] Calculate EOQ for the TVS.

Problem No. 4
A retailer needs to choose between two suppliers for one of its products. The only criterion used for the decision is the cost.  The following information about the product is available:

Demand 200 a week
Ordering cost (for all suppliers) \$75 per order
Holding cost 20% of the unit cost
Working weeks 50 a year

The retailer has narrowed down the choices to two suppliers. The following shows the price-break schedule for each supplier:
SUPPLIER A   SUPPLIER B
Quantity Unit Price   Quantity Unit Price
1-299 \$14.00    1-249 \$14.10
300-699 13.80    250-449 13.90
700+ 13.60    450+ 13.70

A. [4 points] Which supplier should the retailer choose? Explain and support your answer with appropriate calculations. Do NOT exceed the box for explanation.
B. [3 points] What is the optimal lot size for the item? (Note: There has to be only ONE choice, regardless of supplier).
C. [3 points] What is the total annual cost of inventory for the chosen (best) lot size?

Problem No. 5

A retailer is considering a P-system of inventory control for one of its products. However, the total cost of the system (including safety stock) is a concern. The following information about this item is gathered:

Average demand for the product 120 units per day
Standard deviation of demand 30
The store operates 300 days a year
Holding cost 35 percent of the unit cost
Ordering costs \$120 per order
Production lead time (setup time) 3 days
Stock out policy No more than 7%
Unit cost \$9.25

Given this information, determine the following:

A. [5 points] Assuming a P system with a review period of 14 days, calculate the safety stock needed to support the desired stockout policy.
B. [5 points] Annual total cost of the P-system (Grand total cost of inventory for the year)

Problem No. 6

[5 points] A retailer currently holds 45 units of safety stock for one of its products. Demand for this product averages 100 units a week with the standard deviation of 12 and the lead time for it is 3 weeks. The current industry standard for this firm is 2% stockout. Is this retailer competitive? WHY? You must show your calculation and explanation below, otherwise there will be no credit.
SHOW BOTH YOUR KEY FIGURES AND YOUR EXPLANATION IN THE FIELD FOR THIS QUESTION

BONUS PROBLEM # 2

[5 points] A retailer is considering a P-system of inventory control for one of its products. However, the first question is what the review period (P) should be. The following information about this item is gathered:

Average demand for the product 150 units per day
The store operates 300 days a year
Holding cost 30 percent of the unit cost
Ordering cost \$90 per order
Unit cost \$10.75

Given this information, determine the best estimate for the P (Review Period)

CUMULATIVE STANDARD NORMAL DISTRIBUTION TABLE (z-TABLE)

Z 0 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 0.5753
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 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 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 0.7995 0.8023 0.8051 0.8078 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 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 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 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 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 0.9864 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 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 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 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998

For Z = 2.0

The figures in the table show the cumulative area under the curve from minus infinity to a given positive z value

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