Fixed and Variable CostsIn connection with surfacing a new highway, a contractor has a choice of two sites on which to set up the asphalt-mixing plant equipment. The contractor estimates that it will cost $1.15 per cubic yard mile (yd3-mile) to haul the asphalt-paving material from the mixing plant to the job location. Factors relating to the two mixing sites are as follows (Production costs at each sites are the same):Cost factorSite ASite B

Average hauling distance6 miles4.3 miles

Monthly rental of site$1,000$5,000

Cot set up and remove equipment$15,000$25,000

Hauling expense$1.15/yd3-mile$1.15/yd3-mile

FlagpersonNot required$96/day

The job requires 50,000 cubic yards of mixed-asphalt-paving material. It is estimated that four months (17 weeks of five working days per week) will be required for the job. Compare the two sites in terms of their fixed, variable and total costs. Assume that the cost of the return trip is negotiable. Which is the better site? For the selected site, how many cubic yards of paving material does the contractor have to deliver before starting to make a profit if paid $8.05 per cubic yard delivered to the job location?

Optimal Demand When Demand Is a Function of PriceA company produces an electronic timing switch that is used in consumer and commercial products. The fixed cost (CF) is $73,000 per month, and the variable cost (cv) is $83 per unit. The selling price per unit is p = $180 0.02(D), based on equation. For this situation, (a) determine the optimal volume for this product and confirm that a profit occurs (instead of a loss) at this demand; (b) find the volumes at which breakeven occurs; that is, what is the range of profitable demand?

Breakeven Point When Price Is Independent of DemandAn engineering consulting firm measures its output in a standard service hour unit, which is a function of the personnel grades levels in the professional staff. The variable cost (cv) is $62 per standard service hour. The charge-out rate [i.e., selling price (p)] is $85.56 per hour. The maximum output of the firm is 160,000 per year, and its fixed cost (CF) is $2,024,000 per year. For this firm, (a) what is the breakeven point in standard service hours and in percentage of total capacity? (b) What is the percentage reduction in the breakeven point (sensitivity) if fixed costs are reduced 10%; if variable cost per hour is reduced 10%; and if the selling price per unit is increased by 10%?

How Fast Should the Airplane Fly?The cost of operating a jet-powered commercial (passenger-carrying) airplane varies as the three-halves power of its velocity; specifically, CO = knv3/2, when n is the trip length in miles, k is a constant of proportionality, and v is velocity in miles per hour. It is known that at 400 miles per hour, the average cost of operation is $300 per mile. The company that owns the aircraft wants to minimize the cost of operation, but that cost must be balanced against that cost of passengers time (CC), which has been set at $300,000 per hour. (a) At what velocity should the trip be planned to minimize the total cost, which is the sum of the cost of operating the airplane and the cost of passengers time? (b) How do you know that your answer for the problem in Part A minimizes the total cost?

Energy Savings through Increased InsulationThis example deals with a discrete optimization problem of determining the most economical amount of attic insulation for a large single-story home in Virginia. In general, the heat lost through the roof of a single-story home is

In southwest Virginia, the number of heating days per year is approximately 230, and the annual heating degree-days equals 230 (65F-46F) = 4370 degree-days per year. Here, 65F is assumed to be the average inside temperature and 46F is the average outside temperature each day.Consider a 2,400-ft2 single-story house in Blacksburg. They typical annual space-heating load for this size of a house is 100 x 106 BTU. That is, with no insulation in the attic, we lose about 100 x 106 BTE per year.* Common sense dictates that the no insulation alternative is not attractive and is to be avoided.With insulation in the attic, the amount of heat lost each year will be reduced. The value of energy savings that results from adding insulation and reducing heat loss is dependent on what type of residential heating furnace is installed. For this example, we assume that an electrical resistance furnace is installed by the builder, and its efficiency is near 100%.Now were in a position to answer the following question: What amount of insulation is most economical? An additional piece of data we need to involves the cost of electricity, which is $0.0074 per kWh. This can be converted to dollars per 10^6 BTU as follows (1kWh = 3,413 Btu):

The cost of several insulation alternatives and associated space-heating loads for this house are given in the following tableAmount of Insulation

R11R19R30R38

Investment cost ($)60090013001600

Annual heating load (Btu/year)74x10669.8x10667.2x10666.2x106

In view of these data, which amount of attic insulation is most economical? The life of the insulation is estimated to be 25 years.

To Produce or Not to Produce?A manufacturing plant consists of three departments: A, B, and C. Department A occupies 100 square meters in one corner of the plant. Product X is one of the several products being produced in Department A. The daily production of Product X is 576 pieces. The cost accounting records show the following average daily production costs for Product X:

The department foreman has recently learned about an outside company sells X at $0.35 per piece. Accordingly, the foreman figured a cost per day of $0.35(576) = $201.60, resulting in a daily savings of $288.40-$201.60=$86.80. Therefore, a proposal was submitted to the plant manager for shutting down the production line of Product X and buying it from the outside company.However, after examining each component separately, the plant manager decided not to accept the foremans proposal based on the unit cost of product X:1. Direct labor: because the foreman was supervising the manufacture of the other products in Department A in addition to Product X, the only possible savings in labor wound occur if the operator working 4 hours per days on Product X were not reassigned after this line is shut down. That is, a maximum savings of $90.00 per day would result.2. Materials: The maximum savings on direct material will be $86.40. However, this figure could be lower if some of the material for Product X is obtained from scrap of another product.3. Overhead: Because other products are made in Department A, no reduction in total floor space requirements will probably occur. Therefore, no reduction in overhead costs will result from discontinuing Product X. It has been estimated that there will be daily savings in the variable overhead costs traceable to Product X of about $3.00 due to a reduction in power costs and in insurance premiums.

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