Estimating Heating Costs and Oil Delivery Schedule Developed by: L. Carl Leinbach, Gettsburg College

Determining a Delivery Schedule for Heating Oil A heating oil firm gets a new customer on January 11. The firm’s policy is to deliver a full tank of oil (250 gal) and then refill the tank 10 days later. By noting the daily temperature fluctuations and the amount of oil required by the customer on the first fill-up, the firm can work out a delivery schedule for the driver who will be servicing the customer. The firm wants to schedule the deliveries so that the customer’s tank never drops below 50 gallons and also so the driver is not making any unnecessary trips to merely top off the customer’s tank. The delivery schedule is to be made up through the end of March. The heating-industry standard for determining the heating requirements of a site is the number of gallons of fuel used per Heating Degree Day. This method of estimation has proven to be extremely accurate. Heating Degree Days are determined by the following formula. ( Daily High Temp + Daily Low Temp) 2 Only days for which Heating Degree Days is positive will contribute to the heating oil consumption. Table 1 is a record of the temperature data for the 10-day period between deliveries. 65 !

When the oil delivery is made on January 21, the customer requires 167 gallons of fuel. This, coupled with the temperature data, give the firm a way of figuring the heating requirements of the customer in terms of gallons of fuel per Heating Degree Day (HDD).

gal

HDD

=

Fuel Consumed During Time Period 167 = ! 0.444 Total Heating Degree Days in Period 376

From this information, the firm then determines the next delivery date, so that

0.444 • Accumulated HDD = 200

Continued . . .

We must now find a way to determine the accumulated HDD during the time period in question. To do this, we will need to find a function for the HDD and use the integral to determine the accumulation. Daily Temperature Date January January January January January January January January January January

11 12 13 14 15 16 17 18 19 20

Day of Year 11 12 13 14 15 16 17 18 19 20

High 41 38 35 36 35 36 34 38 36 35

Low 20 22 17 20 10 18 20 18 22 19

Degree Days 34.5 35 39 37 42.5 38 38 38 36 38

Table 1: Temperature Data for 1/11-1/20

To find a function that estimates the HDD, the average yearly temperature data are given in Table 2 and a graphical display of this data given in Figure 1. Daily Temperature

Date January February March April May June July August September October November December

15 15 15 15 15 15 15 15 15 15 15 15

Day of Year 15 46 74 105 135 166 196 227 258 288 319 350

High 36 39 49 63 73 82 86 85 78 66 53 40

Low 22 23 31 41 51 61 65 65 57 45 36 26

Degree Days 36 34 25 13 3 -6.5 -10.5 -10 2.5 9.5 20.5 32

Table 2: Average Temperature Data for 1/11-1/20

The graph of the data in the table has the appearance of a sine curve. Certainly, it should come as no surprise that temperature data is periodic. We expect the temperatures during certain times of the year to be the same as at the same time during preceding years. If the temperatures vary greatly from the expected norm, it is a newsworthy event.

Figure 1: A Graphic Display of the Data From Table 2

Continued . . .

Thus, we are going to guess that the function that describes the average HDD function is a sine function with period 365, i.e.,

& 2( # HDD(t ) = A + B • sin $ (t ' c) ! . % 365 " We need to determine the values for A, B, and C to complete our analysis. Examining the data and the graph of the data leads us to the following analysis. The maximum value for HDD is 36, and its minimum is –10.5. Thus, the amplitude of the sine function is approximately 23. If this is the case, then the graph of the function will be centered about the line y = 13. The curve crosses this line in an upward sweep on approximately day 300. Therefore, our initial estimate for HDD is:

& 2( # HDD(t ) = 13 + 23 • sin $ (t ' 300) ! % 365 " A graph of this function is displayed in Figure 2.

Figure 2: First Attempt to Find a Continuous Curve to Fit the Data

Figure 3: Second Attempt to Find a Continuous Curve to Fit the Data

The graph (Figure 2) follows the general shape of the data, but could be improved by shifting it to the left and slightly downward. Experimenting with some modifications of the function, the following function seems to give a fairly close fit to the data in Table 2.

& 2( # HDD(t ) = 12.5 + 23 • sin $ (t ' 290) ! % 365 " The graph of this function and the original data is shown in Figure 3. In order to determine the next delivery date for the customer, we must find an X such that:

.

X

21

& 200 , 2/ )# $12.5 + 23 sin * 365 (t - 290) '! dt = 0.444 = 450.45 + (" %

Continued . . .

This equation takes into account the fact that the last delivery was made on day 21, the estimated fuel consumption is 0.444 gallons per Heating Degree Day, and the firmâ€™s desire to make a delivery of approximately 200 gallons of fuel. Using Mathematica to find the solution to the above equation, yields the solution, X ! 33.859223379 (I used FindRoot). Thus, the next delivery should be on day 34 or February 3. To find the next delivery date, solve the equation X) 200 # 2! &, / 34 +*12.5 + 23sin %$ 365 (t " 290)(' .- dt = 0.444 = 450.45 and continue in this manner, substituting the new delivery date for the lower limit of the integral, until X exceeds 90, i.e., the next delivery date is past March 31. The delivery schedule for the customer is: February 3, February 16, March 3 and March 20.

Your Report: 1. Draw the graph of the data from Table 2, together with the better fitting HDD function. 2. Verify the computations which give the delivery dates of February 16, March 3 and March 20. 3. If oil is delivered on April 13 (day 103), find the date of the next oil delivery. BE CAREFUL! When you make this calculation you may have to consider that the customer will not use any heating oil during the summer months. 4. On the average, how many gallons of fuel can the customer expect to use in a normal year? Note again: Negative HDD do not contribute to fuel consumption. 5. Assume that the temperatures from February 16 to March 31 are 10 degrees warmer than usual (they return to normal after March 31), and that the company keeps to its policy of delivering as close to 200 gallons as possible. When will the next delivery after February 16 take place? When you do this, draw a new graph of the Heating Degree Days, with the higher temperatures from February 16 through March 31 reflected. 6. The customer has asked to be placed on the budget plan, i.e., to pay a fixed monthly amount based on estimated annual fuel consumption. If the fuel costs an average of $1.75 per gallon, determine the customerâ€™s monthly bill based on the estimate of yearly consumption. Again, note that negative Heating Degree Days do not contribute to fuel consumption. End

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