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Buying and Selling

Problems and tasks from a variety of sources intended to illustrate the way mathematics arises in life and work.



Selling Tickets. Many customers of a small Museum near Disney World pay in foreign currency, or with discount coupons from nearby hotels, or with "DisneyDollars"--certificates used to promote local businesses. The Museum offers adult and children's tickets, as well as multi-day passes that cost 50% more than a one-day ticket. The museum also makes allowance for occasional refunds, for complimentary tickets, and for exchanges of one kind of ticket for another. As manager, you need to set up a spread sheet that includes cells for all the variety of ticket sales that may occur and that keeps a running daily balance of the value of tickets sold, as well as weekly and monthly totals.

Concession Business. Given weekly data for one summer on the level of traffic (busy, steady, slow), revenue generated, and hours logged by nine workers who operate mobile concession stands at an amusement park, determine which of these workers to hire for the next summer.

Harvesting Trees. Using data on the growth pattern of trees, a forester must decide each year which of his trees (small, medium, large) he should harvest for a local paper mill. Examine various harvest strategies in terms of short-term and long-range impact.

Managing Inventory. A recent strike at a small firm that manufactures brake linings threatened to stop a significant fraction of automotive production for lack of an inventory of parts. Indeed, the rise of efficient package delivery services and instantaneous computer communication has led many firms to operate with minimum inventories, thus saving warehousing costs but risking shutdown if any part of the network of suppliers fails. Analysis of this situation leads quickly to issues of capacity, redundancy, single-point failures, and time of delivery.

Selling Produce. The proprietor of a busy farmer's market needs to weigh or count produce (depending on what it is), calculate prices based on weight or counts, keep a running total, and make change-- and at the same time bag produce. Efficient cashiers typically use a combination of calculator, paper-and-pencil, and mental arithmetic: calculators to multiply, paper to record items and total the bill, and mental arithmetic for last minute additions.

Scheduling Deliveries. A van drops off and picks up packages for local businesses, some of which have very tight deadlines. In addition, some deliveries are rather bulky, requiring a great deal of space. The driver begins the day with a list of pickups scheduled for that day; throughout the day a dispatcher relays additional requests for pickups. The driver has to plan the order of pickups and deliveries, taking into account (a) their priority; (b) their locations; (c) available space in the van; (d) new requests for pickup; and (e) traffic congestion. Late deliveries are bad for business, but excessive backtracking and speeding tickets are also costly.

Loading a Truck. A truck driver for a local appliance company has to plan the day's schedule and load a van whose cargo space measures 4'3" x 5'6" x 7'. The day's deliveries include four refrigerators, each in a 34" x 34" x 68" carton; three stoves, two in 32" x 28" x 50" cartons and one in a 32" x 28" x 74" carton; and a large freezer in an 80" x 30" x 30" carton. The driver also has to pick up two television sets and a washing machine that have to be repaired. Before beginning, the driver will use a map of the delivery area to plan the day's deliveries in order to notify each customer of the estimated delivery times.

Shipping Products. A local company that makes small appliances needs to manage the shipment of its products. This is not a simple task. Depending on size, destination, and urgency, a shipment may be made by mail, by UPS, or by freight. Each option has weight and size limits; freight only makes sense when the weight is over a certain threshold, whereas mail and UPS will only handle packages that are under certain limits. In addition, UPS and the freight companies often offer deals for bulk shipments or for volume use over a specified time period.

Launching a New Venture. A food company processes cereals and grains into a wide variety of products, most of which are packaged in pre-printed lightweight cardboard boxes. Because of the length of time required to procure new packages, the questionable reliability of key subcontractors, and the availability of desktop printing for package design, the company has decided to prepare a business plan for "insourcing" package manufacturing. This will involve consideration of available (or new) floor space, the cost of new equipment, available vs. new employees, retraining, overtime, and supplier contracts.

Managing a Store. A shoe store in a large shopping mall regularly faces important business decisions caused by the way the mall works. These include decisions about matching hours with the anchor department stores, matching sales of the mall's other shoe stores, as well as deciding about location (near or distant from other shoe stores or the anchor stores). Records of business and sales help the manager make these decisions, but only if carefully maintained and analyzed. Graphs of data gathered over a period of time can often reveal patterns that override anecdotal details that store employees tend to remember most vividly.

Spreadsheets. Set up and validate a spreadsheet to record expenses and project budget trends for a company that has three products and a small staff.

Flowcharts. Create a flowchart for a manufacturing process that includes appropriate statistical process controls, Pareto analysis, data histograms, and fishbone diagrams.




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Last Update: 12/28/98