OPTIMIZATION USING SPREADSHEET MODELING
Coastal Telephone Company (CTC) is a regional supplier of long-distance telephone services. CTC is trying to determine the optimal pricing structure for its daytime and evening long distance calling rates. The daytime price applies from 8:00 A.M. to 6:00 P.M., and the evening price applies the rest of the time. With the help of a consultant, the company has estimated the average demand for phone lines (per minute) as follows:
Daytime Lines Demanded = 600 - 5000 x Day Price + 300 x Evening Price
Evening Lines Demanded = 400 + 600 x Day Price - 2500 x Evening Price
CTC wants to find prices that maximize its revenue. 28044
The Woodstock Appliance Company carries four products. The annual demands for these products range from 300/year for a high-end vacuum cleaner to 30,000/year for a table fan. The order cost, holding cost, and purchase cost are known for each of the four products as well as how much space each product occupies. The numerical information is summarized in the following table:
Woodstock stores inventory in its warehouse, which contains 12,000 square feet that can be dedicated to any combination of the four products. The problem is to find the order quantities that minimize cost while respecting the limit on storage space. 43916
Kilroy Paper Company distributes specialty papers to big-box stores in ten major U.S. metropolitan areas and plans to consolidate its warehouses into one national distribution center (DC). To identify a suitable location, Kilroy's distribution manager first maps the ten stores on a two-dimensional grid, so that coordinates (xk, yk) can be associated with each site. These values are shown in the following table:
For any distribution center location (x, y), it is possible to calculate the distance from the DC to each of the stores and to sum the distances. This total can be thought of as a proxy for the total annual cost incurred by Kilroy's trucks, since they will make regular trips to the individual stores. Minimizing the sum of distances therefore represents an objective that is consistent with minimizing annual distribution cost. Kilroy wishes to determine the location that achieves the minimum sum of distances. 670.6
The Diaz Coffee Company blends three types of coffee beans (Brazilian, Colombian, and Peruvian) into ground coffee to be sold at retail. Suppose that each kind of bean has a distinctive aroma and strength, and the company has a chief taster who can rate these features on a scale of 1 to 100. The features of the beans are tabulated as follows:
The company would like to create a blend that has an aroma rating of at least 78 and a strength rating of at least 16. Its supplies of the various beans are limited, however. The available quantities are specified above. All beans are delivered under a previously arranged purchase agreement. Diaz wants to make four million pounds of the blend at the lowest possible cost. 2448
Coach Kemppel is the coach of the Buchanan Swim Club's co-ed team. Her team competes against other swim clubs, and a perennial question for the coach is how to organize the medley relay team. The medley relay requires four swimmers to each swim a different stroke: butterfly, breaststroke, backstroke, and freestyle. The relay is the final event in the competitions, and the outcome of the swim meet often depends on the performance of the relay team. During practice, Coach Kemppel has asked each of her top four swimmers to try each of the four strokes, and she has tracked their times (in seconds), as shown in the following table:
With this information, Coach Kemppel is ready to assign swimmers to strokes in the relay race, but she can see that a lot of combinations are possible. 173
The Western Paper Company manufactures paper at two factories (F1 and F2) on the West Coast. Their products are shipped by rail to a pair of depots (D1 and D2), one in the Midwest and one in the South. At the depots, the products are repackaged and sent by truck to three regional warehouses (W1, W2, and W3) around the country, in response to replenishment orders. Each of the factories has a known monthly production capacity, and the three regional warehouses have placed their demands for next month. The following tables summarize the data that have been collected for this planning problem:
Knowing the costs of transporting goods from factories to DCs and from DCs to warehouses, Western Paper is interested in scheduling its material flow at the minimum possible cost. 12881
The Ligon Paper Company specializes in paper recycling. The company owns several facilities that obtain paper from commercial or municipal sources, and they produce paper for a variety of markets where customers are looking for recycled content.
Ligon Paper collects three types of input, which they classify as White Paper, Mixed Paper, and Newsprint. Applying various processes, they produce three products: high-quality Office Paper, lower-quality Catalog Paper, and napkin-grade Tan Stock.
One of the processes at Ligon Paper takes White Paper and converts it to Office Paper. For each ton of White Paper input, the process generates 0.85 ton of Office Paper. Alternatively, a ton of White Paper can be converted to Catalog Paper; here the yield is 0.90 ton. Thirdly, White Paper can be converted to Tan Stock, with a yield of 95 percent. For the other types of input, different yields apply, and it is not possible to convert Newsprint to Office Paper. The following table gives a complete set of yield factors, stated as percentages:
In the coming planning period, Ligon's contracts with suppliers have generated 300 tons of White Paper, 600 tons of Mixed Paper, and 400 tons of Newsprint. In existing markets, Ligon could sell 150 tons of Office Paper at the market price of $25/ton, 750 tons of Catalog Paper at $20/ton, and 550 tons of Tan Stock at $18/ton. The problem is to determine how much of each product to produce. 19814
Parents Patti and Russ want to provide for their daughter's college expenses with some of the $100,000 they have recently inherited. They hope to set aside part of the money and establish an account that would cover the needs of their daughter's college education, which begins four years from now, with a one-time investment. Their estimate is that first-year college expenses will come to $24,000 and will increase $2,000 per year during each of the remaining three years of college. The following investment instruments are available:
Faced with this prospect, Patti and Russ wish to set aside the minimum amount of money initially that will guarantee that the college expenses will be met. In other words, they seek an investment plan that will cover college financial needs with the smallest possible initial investment. 80883
The city of Metropolis is in the process of designing a new public emergency system, and their design calls for locating emergency vehicles around the city. The city is divided into nine districts, and seven potential sites have been identified as possible locations for emergency vehicles. Equipment located at each potential site can reach some (but not all) of the districts within the 3-minute time requirement specified by the city. In the following table, an entry of 1 means that the district can be serviced from the corresponding site within the time requirement.
The city wants to provide coverage to all nine districts within the specified time, using the minimum number of sites. 3
Mayhugh Manufacturing, a medium-size job shop, has been producing and selling three product families. Each product family requires production hours in each of three departments. In addition, each product family requires its own sales force, which must be supported no matter how large or small the sales volume happens to be. The parameters describing the situation are summarized in the following table: 508
Miles Manufacturing Company is a regionally focused production shop that fabricates metal components for auto companies. Its scheduling efforts are centered on a flexible machining center that handles a variety of operations, such as drilling, trimming, polishing, and mechanical testing. Jobs arrive at the machine—each job corresponding to a customer order—and the information system provides data on the size of the order, how long it will take to process and when it is due (the due dates having been previously negotiated with customers.) These due dates, which apply within the production scheduling system, have been adjusted for the delivery time needed to place the order in the customer's hands. When several orders are waiting to be processed, the supervisor looks for guidance on how the jobs should be sequenced. The ideal schedule would allow all jobs to be completed on or before their due dates.
This morning's workload consists of six jobs, as described in the following table. The problem is to sequence the jobs, thereby determining the machine schedule for the next few days.
With 60 total hours of work to schedule, and a latest due date of 40, it is obvious that the jobs cannot all be finished on time, even in the best schedule. Each job will be either on time or late, as a function of the sequence chosen. If it is late, the amount of time by which it misses its due date is called its tardiness. There is no tardiness when a job completes prior to its due date. The objective is to minimize the total tardiness in the schedule. 33
The Douglas Electric Cart Company assembles small electric vehicles that are sold for use on golf courses, at university campuses, and in sports stadiums. In these markets, customers like a variety of colors, so Douglas offers several choices. As a result, its manufacturing operations include a sophisticated painting operation, which is scheduled separately from other manufacturing operations. In today's schedule, there are six colors (C1 through C6) with cleaning times as shown in the following table:
The entry in row i and column j of the table gives the cleaning time required between batches of color Ci and batches of color Cj. Each production run consists of a cycle through the full set of colors, and the operations manager wishes to sequence the colors so that the total cleaning time in a cycle is minimized. 167
Cutting Stock. Poly Products sells packaging tape to industrial customers. All tape is sold in 100-foot rolls that are cut in various widths from a master roll, which is 15 inches wide. The product line consists of the following widths: 200, 300, 500, 700, and 1100. These can be cut in different combinations. For example, one combination might consist of three cuts of 500 each. Another combination might consist of two 200 cuts and an 1100 cut. Both of these configurations use the entire 15-inch roll without any waste, but other configurations are also possible. For example, another combination might consist of two 700 cuts. This combination creates 1 inch of waste for every roll cut this way. Each week, Poly Products collects demands from its customers and distributors and must figure out how to configure the cuts in the master rolls. To do so, the production manager lists all possible combinations of cuts and tries to fit them together so that waste is minimized while demand is met. (In particular, demand must be met exactly, because Poly Products does not keep inventories of its tape.) This week's demands are shown in the following table:
How many configurations can be cut from a 15-inch master roll so that there is less than 2 inches of waste (i.e., the smallest quantity that can be sold) left on the roll? What is the minimum amount of waste that can be created if all demand is met exactly?
National Metals Company (NMC) manufactures titanium shafts. Its equipment is capable of producing shafts in ten lengths, reflecting different settings on its machinery. These lengths are 32 cm to 50 cm in steps of 2 cm. Setting up the machinery to produce one of these lengths (which is done once a week) costs $250. As a result, NMC has decided to make only a selected number of lengths. When a customer requests a given length, NMC may supply it from stock if it happens to match one of the lengths in the production schedule. Otherwise, NMC trims a longer length to meet the order. The variable cost for producing the shafts is $20 per cm, and NMC receives revenue of $40 per cm. Trim waste can be sold to a recycler for $15 per cm. The demand requirements for the coming week are tabulated as follows—all demand must be satisfied:
What is an optimal assortment of lengths for NMC to manufacture?
Universal Technologies, Inc. has identified two qualified vendors with the capability to supply certain of its electronic components. For the coming year, Universal has estimated its volume requirements for these components and has obtained price-break schedules from each vendor. Universal's engineers have also estimated each vendor's maximum capacity for producing these components, on the basis of available information about equipment in use and labor policies in effect. Finally, because of its limited history with vendor A, Universal has adopted a policy that permits no more than 60 percent of its total unit purchases on these components to come from vendor A.
What is the minimum total cost for Universal's purchases?
In the optimal solution to part (a), which purchases are made at discounted prices?
In the previous problem, suppose that vendor A provides a new price-discount schedule for component 3. This one is an "incremental" discount, as opposed to an "all-units" discount, as follows:
Unit price = $60 on all units up to 1000
Unit price = $56 on the next 1000 units
Unit price = $51 on the next 500 units
a. What is the minimum total cost for Universal's purchases?
b. In the optimal solution to part (a), which purchases are made at discounted prices?