Problem #1
Background: To illustrate the applications of statistics to quality control, we will use an example from a Midwest pharmaceutical company that manufactures individual syringes with a self-contained, single dose of an injectable drug.1 In the manufacturing process, sterile liquid drug is poured into glass syringes and sealed with a rubber stopper.
The remaining stage involves insertion of the cartridge into plastic syringes and the electrical "tacking" of the containment cap at a precisely determined length of the syringe. A cap that is tacked at a shorter-than-desired length (less than 4.891 inches) leads to pressure on the cartridge stopper and, hence, partial or complete activation of the syringe. Such syringes must then be scrapped. If the cap is tacked at a longer-than-desired length (4.959 inches or longer), the tacking is incomplete or inadequate, which can lead to cap loss and potentially a cartridge loss in shipment and handling. Such syringes can be reworked manually to attach the cap at a lower position.
However, this process requires a 100% inspection of the tacked syringes and results in increased cost for the items. This final production step seemed to be producing more and more scrap and reworked syringes over successive weeks. At this point, statistical consultants became involved in an attempt to solve this problem and recommended statistical process control for the purpose of improving the tacking operation.
- the specifications on the syringe length are 4.925±0.034
Build a spreadsheet for computing the Process Capability Index that allows you to enter up to 100 data points and apply it to the third shift syringe data in the Syringe Samples data sheet.
Problem #2
Think of any retailer that operates many stores throughout the country, such as Old Navy, Hallmark Cards, or Radio Shack, to name just a few. The retailer is seeking to open new stores and needs to evaluate the profitability of a proposed location that would be leased for five years. An Excel model is provided in the adjacent New Store Financial Analysis.
Use Scenario Manager to evaluate the cumulative discounted cash flow for the fifth year under the following scenarios:
Problem #3.
Using this model, developed from the model for problem #2, use Solver to determine the optimum mix for the decision variables to maximise Cumulative Discounted Cash Flow. Note: this is not quite a fully linear model, so use the GRG Non-linear Method.
Problem #4
Midwestern Hardware must decide how many snow shovels to order for the coming snow season. Each shovel costs $15.50 and is sold for $29.99. No inventory is carried from one snow season to the next. Shovels unsold after February are sold at a discount price of $8.99.
Past data indicate that sales are highly dependent on the severity of the winter season. Past seasons have been classified as mild, average, or harsh, and the following distribution of regular price demand has been tabulated:
Shovels must be ordered from the manufacturer in lots of 250. Construct a decision tree to illustrate the components of the decision model, and find the optimal quantity for Midwestern to order if the forecast calls for a 30% chance of an average winter and a 60% chance of a harsh winter. [Hint: Develop an Order-Demand matrix showing the profit on each order-demand combination.
Because Decision Tree is limited to 5 initial branches, use order sizes of 1000, 1750, 2500, 2750, and 3000.]
Problem #5
A small factory has two serial work stations: mold/trim and assemble/package. Jobs to the factory arrive at an exponentially distributed rate of 1 every 8 hours. Time at the mold/trim station is exponential, with a mean of 6 hours. Each job proceeds next to the assemble/package station and requires an average of 4 hours, again exponentially distributed. Using SimQuick, estimate the average time in the system, average waiting time at mold/trim, and average waiting time at assembly/packaging.
Problem#6
A department store chain is planning to open a new store. It needs to decide how to allocate the
125,000 square feet if available floor space among seven departments. Data on expected performance of each department per month, in terms if square feet (sf), are shown in the table.
The company has gathered $25 million to invest in floor stock. The risk column is a measure of risk associated with investment in floor stock based on past data from other stores and accounts for outdated inventory, pilferage, breakage, etc. For instance, electronics loses 24% of its total investment; furniture loses 12% of its total investment, etc. The maximum total risk can be no greated than 12% of the actual investment.
a. Develop a linear optimization model to maximize profit.
b. If the chain obtains another $2.5 million of investment capital for stock, what would the new solution be?
Problem#7
Sue Strand manages a professional theater group in a major city. Her marketing plan is focused on generating additional local demand for plays and increasing ticket revenue, and also gaining attention to the national level to build awareness of the theater group across the country.
She has $35,000 to spend on media advertising. The goal of the advertisement campaign is to generate as much local recognition as possible while reaching at least 4,500 units national exposure. She has set a limit of 125 total ads.
Additional information shown in the table to the right. The last column sets limits on the number if ads to ensure that the advertising markets do not become saturated.
a. Find the optimal number of ads of each type to run to meet the choir's goals by developing and solving an integer optimization model.
b. What if she decides to use no more than six different types of ads? Modify the model in part (a)to answer this question.
Attachment:- 7_questions.rar