About Deadhead Calculation Formulas

There are four formulas that Schedule Server can use to calculate trip costs for deadhead:

Formula 1
Formula 2
Formula 3
Formula 4

The following before and after example will be used to illustrate these calculation formulas:

Example when Before trip 3 (PU3/DO3) is inserted

Example when after trip 3 (PU3/DO3) is inserted

Note that the example calculations for each formula do not show that one way of calculating deadhead cost is better than the other. Selecting a formula to be used at a site is dependent on that site’s needs (for example, reducing slack or deadhead time) and should be used with caution. If in doubt, select formula 2, which considers slack and deadhead time without severe penalties.

For the purpose of the examples in this section, the values to be used in each calculation are listed in the following table:

The Deadhead Costing Weight (CWDH) is not measured in minutes or seconds, but its value for these examples will be 10.

Table 1. Example of Deadhead Costing Weight
Variable	Description	Minutes	Seconds (used in calculations)
dht₁	Deadhead Time Before	85	5100
dht₂	Deadhead Time After	50	3000
slack₁	SlackTime Before	5	300
slack₂	Slack Time After	0	0
threshold	Threshold (CWDHTHT)	15	900

Formula 1

This method of calculation is the most sensitive way of determining how deadhead affects a trip’s cost because it relies heavily on the amount of slack available in a run’s itinerary. Its purpose is to incentivize the reduction of deadhead and slack.

The formula for this method is as follows:

To determine the cost of deadhead in a schedule, replace the variables with the corresponding values for each of the “before” and “after” trips that incur deadhead:

Table 2. Cost of a Deadhead Event (Before)
DH Event (Before)	dht₁	Slack₁	Threshold	Formula 1
PO - PU1	600	0	900	-120
DO1 - PU2	2400	300	900	1000
DO2 - PI	1800	0	900	600
Total				1480

Table 3. Cost of a Deadhead Event (After)
DH Event (After)	dht₂	Slack₂	Threshold	Formula 1
PO - PU1	600	0	900	-120
DO1 - PU3	1200	0	900	171.429
DO3 - PI	9	0	900	0
Total				51.429

Finally, to determine the solution's overall cost, multiply the difference between the Before trip cost and the After trip cost by the Minimize Deadhead costing weight (CWDH):

Therefore, calculating the cost of inserting trip 3 (PU3/DO3) using formula 1 is -14285.714.

Formula 2

This method of calculation is similar to the first formula, except that it disregards deadhead events where the deadhead time is equal to or less than the threshold. For best results, Trapeze recommends using this formula for deadhead calculation.

The formula for this method is as follows:

The larger of the two values (0 or the calculated value) is used to complete the calculation of the solution cost.

To determine the cost of deadhead in a schedule, replace the variables with the corresponding values for each of the “before” and “after” trips that incur deadhead:

Table 4. Cost of a Deadhead Event (Before)
DH Event (Before)	dht₁	Slack₁	Threshold	Formula 2
PO - PU1	600	0	900	0
DO1 - PU2	2400	300	900	1000
DO2 - PI	1800	0	900	600
Total				1600

Table 5. Cost of a Deadhead Event (After)
DH Event (After)	dht₂	Slack₂	Threshold	Formula 2
PO - PU1	600	0	900	0
DO1 - PU3	1200	0	900	171.429
DO3 - PI	9	0	900	0
Total				171.429

Finally, to determine the solution’s overall cost, multiply the difference between the Before trip cost and the After trip cost by the Minimize Deadhead costing weight (CWDH):

Therefore, calculating the cost of inserting trip 3 (PU3/DO3) using formula 2 is -14285.714.

Formula 3

This method of calculation is the simplest one to use because it focuses solely on deadhead time. However, using this formula cannot determine whether decreasing deadhead will also reduce slack time. Deadhead is penalized only if deadhead time is greater than the threshold.

The formula for this method is as follows:

The larger of the two values (0 or the calculated value) is used to complete the calculation of the solution cost.

To determine the cost of deadhead in a schedule, replace the variables with the corresponding values for each of the “before” and “after” trips that incur deadhead:

Table 6. Cost of a Deadhead Event (Before)
DH Event (Before)	dht₁	Slack₁	Threshold	Formula 3
PO - PU1	600	0	900	0
DO1 - PU2	2400	300	900	1500
DO2 - PI	1800	0	900	900
Total				2400

Table 7. Cost of a Deadhead Event (After)
DH Event (After)	dht₂	Slack₂	Threshold	Formula 3
PO - PU1	600	0	900	0
DO1 - PU3	1200	0	900	300
DO3 - PI	900	0	900	0
Total				300

Finally, to determine the solution’s overall cost, multiply the difference between the Before trip cost and the After trip cost by the Minimize Deadhead costing weight (CWDH):

Therefore, calculating the cost of inserting trip 3 (PU3/DO3) using formula 3 is -21000.

Formula 4

This method of calculation ignores the amount of deadhead time within a run’s itinerary. It focuses only on slack, and the threshold value determines the minimum amount of slack a site would like to see between two events. Trapeze recommends using one of the other deadhead calculation formulas if deadhead time is an issue at your site.

The formula for this method is as follows:

The larger of the two values (0 or the calculated value) is changed to a negative value and used to complete the calculation of the solution cost.

To determine the cost of deadhead in a schedule, replace the variables with the corresponding values for each of the “before” and “after” trips that incur deadhead:

Table 8. Cost of a Deadhead Event (Before)
DH Event (Before)	dht₁	Slack₁	Threshold
PO - PU1	600	0	900
DO1 - PU2	2400	300	900
DO2 - PI	1800	0	900
Total

Table 9. Cost of a Deadhead Event (After)
DH Event (After)	dht₂	Slack₂	Threshold
PO - PU1	600	0	900
DO1 - PU3	1200	0	900
DO3 - PI	900	0	900
Total

Finally, to determine the solution’s overall cost, multiply the difference between the Before trip cost and the After trip cost by the Minimize Deadhead costing weight (CWDH):

Therefore, calculating the cost of inserting trip 3 (PU3/DO3) using formula 4 is 0.