Chapter 1 Section 4

Excel Companion Chapter 1 section 4

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Addition of Functions	*Highway.xls*

Open the file Highway.xls.

SITUATION: The total cost of maintaining a highway over a long time period consists of two major cost components. The first component is the capital cost of building n stations along the highway where equipment and supplies can be stored and work crews can be located. Cc is directly proportional to the number of stations, n. The higher the number of stations built, the greater the capital cost. Algebraically we can represent the capital cost component as the linear function Cc = A*n where A is a constant and n is the independent variable. The second cost component is associated with daily operations. This includes the cost of moving resources, human and material, from the closest station to different problem sites on the highway in order to carry out functions such as bridge repairs or snow removal. As the number of stations increases, the overall operating costs go down because of factors such as lower average travel times to problem sites. These operating costs, Co, are inversely proportional to the total number of stations. Algebraically we can represent this operating cost component as Co = B/n where B is some constant and n is the number of stations built along the highway. The sum of these two cost components is Ct = A*n + B/n. Although highway authorities must also deal with many other complicated factors (government funding, regulations, labor contracts, municipal boundaries, and changing highway needs) we will focus our attention on the simple relationship between A, B, and the impact of adding stations. Suppose A = $80,000 and B = $600,000 and n = 5 then Cc = 80,000*5 = 400,000, Co = 600,000/5 = 120,000, and Ct = 520,000.

From Table & Charts , we see that building four stations would produce a total cost of $470,00 and building eight stations would produce a total cost of $715,000. We see that the lowest theoretical cost corresponds to building and operating three stations. Each of the graphics below gives us a quick visual representation of the numbers in the table. Table & Charts also shows the two cost components, Cc and Co, in a bar chart. Their sum, Ct, can be inferred from the heights of the stacked bars. The second chart in Table & Charts shows all three functions of the independent variable n. The function for total cost (in green) is a typical example of the sum of two functions.


*Table & Charts*

Explore different values of the parameter A (associated with unit construction costs) and the parameter B (associated with operating costs).

QUESTIONS:

1.	Suppose A = $50,000 and B = $800,000. What is the total cost of building and operating four stations?
2.	Suppose A = $50,000, B = $800,000, What is the greatest number of stations that could be built and operated under a budget cap if:
	a) $500,000 is budgeted for capital costs plus operating costs?
	b) $1,000,000 is budgeted for capital costs plus operating costs?
3.	Suppose A = $50,000 and B = $800,000 then what number of stations corresponds to the minimum total cost?

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