Q1. The Road King Tire Company in Birmingham wants to monitor the quality of the tires it manufactures. Each day the company quality-control manager takes a sample of 100 tires, tests them, and determines the number of defective tires. The results of 20 samples have been recorded as follows:

Sample    Number of Defectives        Sample     Number of Defectives
 1                       14                           11                     18
 2                       12                           12                     10
 3                        9                           13                     19
 4                       10                           14                     20
 5                       11                           15                     17
 6                        7                           16                     18
 7                        8                           17                     18
 8                       14                           18                     22
 9                       16                           19                     24
10                       17                           20                     23

Construct a -chart for this process using limits and describe the variation in the process.

Q2. One of the stages in the process of making denim cloth at the Southern Mills Company is to spin cotton yarn onto spindles for subsequent use in the weaving process. Occasionally the yarn breaks during the spinning process, and an operator ties it back together. Some number of breaks is considered normal; however, too many breaks might mean that the yarn is of poor quality. In order to monitor this process, the quality-control manager randomly selects a spinning machine each hour and checks the number of breaks during a 15-minute period. Following is a summary of the observations for the past 20 hours:

Sample     Number of Breaks           Sample        Number of Breaks
 1                         3                          11                         3
 2                         2                          12                         4
 3                         4                          13                         6
 4                         1                          14                         7
 5                         5                          15                         8
 6                         3                          16                         6
 7                         2                          17                         5
 8                         4                          18                         7
 9                         0                          19                         8
10                         2                          20                         6

Construct a -chart using limits for this process and indicate if the process was out of control at any time.

Q3. The Ambrosia Bakery makes cakes for freezing and subsequent sale. The bakery, which operates five days a week, 52 weeks a year, can produce cakes at the rate of 116 cakes per day. The bakery sets up the cake-production operation and produces until a predetermined number has been produced. When not producing cakes, the bakery uses its personnel and facilities for producing other bakery items. The setup cost for a production run of cakes is $700. The cost of holding frozen cakes in storage is $9 per cake per year. The annual demand for frozen cakes, which is constant over time, is 6000 cakes. Determine the following:

a. Optimal production run quantity (Q)
b. Total annual inventory costs
c. Optimal number of production runs per year
d. Optimal cycle time (time between run starts)
e. Run length in working days.

Q4. Southwood Furniture Company is a U.S.-based furniture manufacturer that offshored all of its actual manufacturing operations to China about a decade ago. It set up a distribution center in Hong Kong from which the company ships its items to the United States on container ships. The company learned early on that it could not rely on local Chinese freight forwarders to arrange for sufficient containers for the company's shipments, so it contracted to purchase containers from a Taiwanese manufacturer and then sell them to shipping companies at the U.S. ports the containers are shipped to. Southwood needs 715 containers each year. It costs $1200 to hold a container at its distribution center, and it costs $6000 to receive an order for the containers. Determine the optimal order size, minimum total annual inventory cost, number of annual orders, and time between orders.